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CONTENTS Foreword i i i Chapter 1 : Introduction 1 Chapter 2 : Collection of Data 9 Chapter 3 : Organisation of Data 22 Chapter 4 : Presentation of Data 40 Chapter 5 : Measures of Central Tendency 58 Chapter 6 : Measures of Dispersion 74 Chapter 7 : Correlation 91 Chapter 8 : Index Numbers 107 Chapter 9 : Use of Statistical Tools 121 APPENDIX A : GLOSSARY OF STATISTICAL TERMS 131 APPENDIX B : TABLE OF TWO-DIGIT RANDOM NUMBERS 134 CHA P T ER 1 Introduction told this subject is mainly around Studying this chapter should enable you to: what Alfred Marshall (one of the • know what the subject of founders of modern economics) called economics is about; “the study of man in the ordinary • understand how economics is business of life”. Let us understand linked with the study of economic activities in consumption, what that means. production and distribution; When you buy goods (you may • understand why knowledge of want to satisfy your own personal statistics can help in describing needs or those of your family or those consumption, production and distribution; of any other person to whom you want • learn about some uses of to make a gift) you are called statistics in the understanding of a consumer. economic activities. When you sell goods to make a profit for yourself (you may be . 1 WHY ECONOMICS? a shopkeeper), you are called a seller. You have, perhaps, already had When you produce goods (you may Economics as a subject for your earlier be a farmer or a manufacturer), you classes at school. You might have been are called a producer. 2 STATISTICS FOR ECONOMICS When you are in a job, working for In real life we cannot be as lucky some other person, and you get paid as Aladdin. Though, like him we have for it (you may be employed by unlimited wants, we do not have a somebody who pays you wages or a magic lamp. Take, for example, the salary), you are called a service- pocket money that you get to spend. holder. If you had more of it then you could When you provide some kind of have purchased almost all the things service to others for a payment (you you wanted. But since your pocket may be a lawyer or a doctor or a money is limited, you have to choose banker or a taxi driver or a transporter only those things that you want the of goods), you are called a service- most. This is a basic teaching of provider. Economics. In all these cases you will be called gainfully employed in an economic Activities activity. Economic activities are ones • Can you think for yourself of that are undertaken for a monetary some other examples where a gain. This is what economists mean person with a given income has by ordinary business of life. to choose which things and in what quantities he or she can Activities buy at the prices that are being • List different activities of the charged (called the current members of your family. Would prices)? you call them economic • What will happen if the current activities? Give reasons. prices go up? • Do you consider yourself a Scarcity is the root of all economic consumer? Why? problems. Had there been no scarcity, there would have been no economic We cannot get something for problem. And you would not have nothing studied Economics either. In our daily If you ever heard the story of Aladdin life, we face various forms of scarcity. and his Magic Lamp, you would agree The long queues at railway booking that Aladdin was a lucky guy. counters, crowded buses and trains, Whenever and whatever he wanted, he shortage of essential commodities, the just had to rub his magic lamp on rush to get a ticket to watch a new when a genie appeared to fulfill his film, etc., are all manifestations of wish. When he wanted a palace to live scarcity. We face scarcity because the in, the genie instantly made one for things that satisfy our wants are him. When he wanted expensive gifts limited in availability. Can you think to bring to the king when asking for of some more instances of scarcity? his daughter’s hand, he got them at The resources which the producers the bat of an eyelid. have are limited and also have INTRODUCTION 3 alternative uses. Take the case of food activities of various kinds. For this, that you eat every day. It satisfies your you need to know reliable facts about want of nourishment. Farmers all the diverse economic activities like employed in agriculture raise crops production, consumption and that produce your food. At any point distribution. Economics is often of time, the resources in agriculture discussed in three parts: consum- like land, labour, water, fertiliser, etc., ption, production and distribution. are given. All these resources have We want to know how the alternative uses. The same resources consumer decides, given his income can be used in the production of non- and many alternative goods to choose food crops such as rubber, cotton, jute from, what to buy when he knows the etc. Thus alternative uses of resources prices. This is the study of Consum- give rise to the problem of choice ption. between different commodities that We also want to know how the can be produced by those resources. producer, similarly, chooses what to Activities produce for the market when he knows the costs and prices. This is the • Identify your wants. How many study of Production. of them can you fulfill? How many of them are unfulfilled? Finally, we want to know how the Why you are unable to fulfill national income or the total income them? arising from what has been produced • What are the different kinds of in the country (called the Gross scarcity that you face in your Domestic Product or GDP) is daily life? Identify their causes. distributed through wages (and salaries), profits and interest (We will Consumption, Production and leave aside here income from Distribution international trade and investment). If you thought about it, you might This is study of Distribution. have realised that Economics involves Besides these three conventional the study of man engaged in economic divisions of the study of Economics about which we want to know all the facts, modern economics has to include some of the basic problems facing the country for special studies. For example, you might want to know why or to what extent some households in our society have the capacity to earn much more than others. You may want to know how many people in the country are really 4 STATISTICS FOR ECONOMICS poor, how many are middle-class, how of numbers relating to selected facts many are relatively rich and so on. You in a systematic form) to be added to may want to know how many are all modern courses of modern illiterate, who will not get jobs, economics. requiring education, how many are Would you now agree with the highly educated and will have the best following definition of economics that job opportunities and so on. In other many economists use? words, you may want to know more “Economics is the study of how facts in terms of numbers that would people and society choose to answer questions about poverty and employ scarce resources that could disparity in society. If you do not like have alternative uses in order to the continuance of poverty and gross produce various commodities that disparity and want to do something satisfy their wants and to about the ills of society you will need distribute them for consumption to know the facts about all these among various persons and groups things before you can ask for in society.” appropriate actions by the government. If you know the facts it Activity may also be possible to plan your own • Would you say, in the light of the life better. Similarly, you hear of — discussion above, that this some of you may even have definition used to be given seems experienced disasters like Tsunami, a little inadequate now? What earthquakes, the bird flu — dangers does it miss out? threatening our country and so on that affect man’s ‘ordinary business 2. STATISTICS IN ECONOMICS of life’ enormously. Economists can look at these things provided they In the previous section you were told know how to collect and put together about certain special studies that the facts about what these disasters concern the basic problems facing a cost systematically and correctly. You country. These studies required that may perhaps think about it and ask we know more about economic facts yourselves whether it is right that in terms of numbers. Such economic modern economics now includes facts are also known as data. learning the basic skills involved in The purpose of collecting data making useful studies for measuring about these economic problems is to poverty, how incomes are distributed, understand and explain these how earning opportunities are related problems in terms of the various to your education, how environmental causes behind them. In other words, disasters affect our lives and so on? we try to analyse them. For example, Obviously, if you think along these when we analyse the hardships of lines, you will also appreciate why we poverty, we try to explain it in terms needed Statistics (which is the study of the various factors such as INTRODUCTION 5 unemployment, low productivity of By data or statistics, we mean both people, backward technology, etc. quantitative and qualitative facts that But, what purpose does the are used in Economics. For example, analysis of poverty serve unless we are a statement in Economics like “the able to find ways to mitigate it. We production of rice in India has may, therefore, also try to find those increased from 39.58 million tonnes measures that help solve an economic in 1974–75 to 58.64 million tonnes in problem. In Economics, such 1984–85”, is a quantitative fact. The measures are known as policies. numerical figures such as ‘39.58 So, do you realise, then, that no million tonnes’ and ‘58.64 million analysis of a problem would be tonnes’ are statistics of the possible without the availability of production of rice in India for data on various factors underlying an 1974–75 and 1984–85 respectively. economic problem? And, that, in such In addition to the quantitative a situation, no policies can be data, Economics also uses qualitative formulated to solve it. If yes, then you data. The chief characteristic of such have, to a large extent, understood the information is that they describe basic relationship between Economics attributes of a single person or a group and Statistics. of persons that is important to record 3. WHAT IS STATISTICS? as accurately as possible even though they cannot be measured in At this stage you are probably ready quantitative terms. Take, for example, to know more about Statistics. You “gender” that distinguishes a person might very well want to know what the as man/woman or boy/girl. It is often subject “Statistics” is all about. What possible (and useful) to state the are its specific uses in Economics? information about an attribute of a Does it have any other meaning? Let person in terms of degrees (like better/ us see how we can answer these questions to get closer to the subject. worse; sick/ healthy/ more healthy; In our daily language the word unskilled/ skilled/ highly skilled etc.). ‘Statistics’ is used in two distinct Such qualitative information or senses: singular and plural. In the statistics is often used in Economics plural sense, ‘statistics’ means and other social sciences and ‘numerical facts systematically collected and stored systematically collected’ as described by Oxford like quantitative information (on Dictionary. Thus, the simple meaning prices, incomes, taxes paid etc.), of statistics in plural sense is data. whether for a single person or a group Do you know that the term statistics of persons. in singular means the ‘science of You will study in the subsequent collecting, classifying and using chapters that statistics involves statistics’ or a ‘statistical fact’. collection and organisation of data. The next step is to present the data in 6 STATISTICS FOR ECONOMICS tabular, diagrammatic and graphic a statistical data. Whereas, saying forms. The data, then, is summarised hundreds of people died, is not. by calculating various numerical Statistics also helps in condensing indices such as mean, variance, the mass of data into a few numerical standard deviation etc. that represent measures (such as mean, variance the broad characteristics of the etc., about which you will learn later). collected set of information. These numerical measures help summarise data. For example, it Activities would be impossible for you to • Think of two examples of remember the incomes of all the qualitative and quantitative data. people in a data if the number of • Which of the following would give people is very large. Yet, one can you qualitative data; beauty, remember easily a summary figure like intelligence, income earned, the average income that is obtained marks in a subject, ability to statistically. In this way, Statistics sing, learning skills? summarises and presents a meaningful overall information about 4. WHAT STATISTICS DOES? a mass of data. Quite often, Statistics is used in By now, you know that Statistics is an indispensable tool for an economist finding relationships between different that helps him to understand an economic factors. An economist may economic problem. Using its various be interested in finding out what methods, effort is made to find the happens to the demand for a causes behind it with the help of the commodity when its price increases qualitative and the quantitative facts or decreases? Or, would the supply of of the economic problem. Once the a commodity be affected by the causes of the problem are identified, changes in its own price? Or, would it is easier to formulate certain policies the consumption expenditure increase to tackle it. when the average income increases? But there is more to Statistics. It Or, what happens to the general price enables an economist to present level when the government economic facts in a precise and expenditure increases? Such ques- definite form that helps in proper tions can only be answered if any comprehension of what is stated. relationship exists between the When economic facts are expressed in various economic factors that have statistical terms, they become exact. been stated above. Whether such Exact facts are more convincing than relationships exist or not can be easily vague statements. For instance, verified by applying statistical saying that with precise figures, 310 methods to their data. In some cases people died in the recent earthquake the economist might assume certain in Kashmir, is more factual and, thus, relationships between them and like INTRODUCTION 7 to test whether the assumption she/ consumption of past years or of recent he made about the relationship is valid years obtained by surveys. Thus, or not. The economist can do this only statistical methods help formulate by using statistical techniques. appropriate economic policies that In another instance, the economist solve economic problems. might be interested in predicting the changes in one economic factor due 5. CONCLUSION to the changes in another factor. For example, she/he might be interested Today, we increasingly use Statistics in knowing the impact of today’s to analyse serious economic problems investment on the national income in such as rising prices, growing future. Such an exercise cannot be population, unemployment, poverty undertaken without the knowledge of etc., to find measures that can solve Statistics. such problems. Further it also helps Sometimes, formulation of plans evaluate the impact of such policies and policies requires the knowledge in solving the economic problems. For of future trends. For example, an example, it can be ascertained easily Statistical methods are no substitute for common sense! There is an interesting story which is told to make fun of statistics. It is said that a family of four persons (husband, wife and two children) once set out to cross a river. The father knew the average depth of the river. So he calculated the average height of his family members. Since the average height of his family members was greater than the average depth of the river, he thought they could cross safely. Consequently some members of the family (children) drowned while crossing the river. Does the fault lie with the statistical method of calculating averages or with the misuse of the averages? economic planner has to decide in using statistical techniques whether 2005 how much the economy should the policy of family planning is produce in 2010. In other words, one effective in checking the problem of must know what could be the ever-growing population. expected level of consumption in 2010 In economic policies, Statistics in order to decide the production plan plays a vital role in decision making. of the economy for 2010. In this For example, in the present time of situation, one might make subjective rising global oil prices, it might be judgement based on the guess about necessary to decide how much oil consumption in 2010. Alternatively, India should import in 2010. The one might use statistical tools to decision to import would depend on predict consumption in 2010. That the expected domestic production of could be based on the data of oil and the likely demand for oil in 8 STATISTICS FOR ECONOMICS 2010. Without the use of Statistics, it cannot be made unless we know the cannot be determined what the actual requirement of oil. This vital expected domestic production of oil information that help make the and the likely demand for oil would decision to import oil can only be be. Thus, the decision to import oil obtained statistically. Recap • Our wants are unlimited but the resources used in the production of goods that satisfy our wants are limited and scarce. Scarcity is the root of all economic problems. • Resources have alternative uses. • Purchase of goods by consumers to satisfy their various needs is Consumption. • Manufacture of goods by producers for the market is Production. • Division of the national income into wages, profits, rents and interests is Distribution. • Statistics finds economic relationships using data and verifies them. • Statistical tools are used in prediction of future trends. • Statistical methods help analyse economic problems and formulate policies to solve them. EXERCISES . 1 Mark the following statements as true or false. i) ( Statistics can only deal with quantitative data. i) (i Statistics solves economic problems. (iii) Statistics is of no use to Economics without data. . 2 Make a list of activities that constitute the ordinary business of life. Are these economic activities? . 3 ‘The Government and policy makers use statistical data to formulate suitable policies of economic development’. Illustrate with two examples. . 4 You have unlimited wants and limited resources to satisfy them. Explain by giving two examples. . 5 How will you choose the wants to be satisfied? . 6 What are your reasons for studying Economics? . 7 Statistical methods are no substitute for common sense. Comment. CHAPTER 2 Collection of Data chapter, you will study the sources of Studying this chapter should enable data and the mode of data collection. you to: • understand the meaning and The purpose of collection of data is to purpose of data collection; collect evidence for reaching a sound • distinguish between primary and and clear solution to a problem. secondary sources; In economics, you often come • know the mode of collection of data; across a statement like, • distinguish between Census and “After many fluctuations the output Sample Surveys; of food grains rose to 176 million tonnes • be familiar with the techniques of sampling; in 1990–91 and 199 million tonnes in • know about some important 1996–97, but fell to 194 million tonnes sources of secondary data. in 1997–98. Production of food grains then rose continuously and touched 212 million tonnes in 2001–02.” 1. I N T R O D U C T I O N In this statement, you can observe In the previous chapter, you have read that the food grains production in about what is economics. You also different years does not remain the studied about the role and importance same. It varies from year to year and of statistics in economics. In this from crop to crop. As these values 1 0 STATISTICS FOR ECONOMICS vary, they are called variable. The 2. WHAT ARE THE SOURCES OF DATA? variables are generally represented by Statistical data can be obtained from the letters X, Y or Z. The values of two sources. The enumerator (person these variables are the observation. who collects the data) may collect the For example, suppose the food grain data by conducting an enquiry or an production in India varies between investigation. Such data are called 100 million tonnes in 1970–71 to 220 Primary Data, as they are based on million tonnes in 2001–02 as shown first hand information. Suppose, you in the following table. The years are want to know about the popularity of represented by variable X and the a film star among school students. For production of food grain in India (in this, you will have to enquire from a million tonnes) is represented by large number of school students, by variable Y: asking questions from them to collect TABLE 2.1 the desired information. The data you Production of Food Grain in India get, is an example of primary data. (Million Tonnes) If the data have been collected and X Y processed (scrutinised and tabulated) 1970–71 108 by some other agency, they are called 1978–79 132 Secondary Data. Generally, the 1979–80 108 published data are secondary data. 1990–91 176 They can be obtained either from 1996–97 199 published sources or from any other 1997–98 194 source, for example, a web site. Thus, 2001–02 212 the data are primary to the source that collects and processes them for the Here, these values of the variables first time and secondary for all sources X and Y are the ‘data’, from which we that later use such data. Use of can obtain information about the secondary data saves time and cost. trend of the production of food grains For example, after collecting the data in India. To know the fluctuations in on the popularity of the film star the output of food grains, we need the among students, you publish a report. ‘data’ on the production of food grains If somebody uses the data collected in India. ‘Data’ is a tool, which helps by you for a similar study, it becomes in understanding problems by secondary data. providing information. You must be wondering where do 3. HOW DO WE COLLECT THE DATA? ‘data’ come from and how do we collect these? In the following sections we will Do you know how a manufacturer discuss the types of data, method and decides about a product or how a instruments of data collection and political party decides about a sources of obtaining data. candidate? They conduct a survey by COLLECTION OF DATA 1 1 asking questions about a particular Good Q product or candidate from a large ) i ( Is the electricity supply in your group of people. The purpose of locality regular? surveys is to describe some ii () Is increase in electricity charges characteristics like price, quality, justified? usefulness (in case of the product) and • The questions should be precise popularity, honesty, loyalty (in case and clear. For example, of the candidate). The purpose of the Poor Q survey is to collect data. Survey is a What percentage of your income do method of gathering information from you spend on clothing in order to look individuals. presentable? Preparation of Instrument Good Q What percentage of your income do The most common type of instrument you spend on clothing? used in surveys is questionnaire/ interview schedule. The questionnaire • The questions should not be is either self administered by the ambiguous, to enable the respon- respondent or administered by the dents to answer quickly, correctly researcher (enumerator) or trained and clearly. For example: investigator. While preparing the Poor Q questionnaire/interview schedule, you Do you spend a lot of money on books should keep in mind the following in a month? points; Good Q How much do you spend on books in • The questionnaire should not be too a month? long. The number of questions i) ( Less than Rs 200 should be as minimum as possible. ii () Between Rs 200–300 Long questionnaires discourage ii (i) Between Rs 300–400 people from completing them. i) (v More than Rs 400 • The series of questions should move • The question should not use double from general to specific. The negatives. The questions starting questionnaire should start from with “Wouldn’t you” or “Don’t you” general questions and proceed to should be avoided, as they may more specific ones. This helps the lead to biased responses. For respondents feel comfortable. For example: example: Poor Q Poor Q Don’t you think smoking should be i) ( Is increase in electricity charges prohibited? justified? Good Q ii () Is the electricity supply in your Do you think smoking should be locality regular? prohibited? 1 2 STATISTICS FOR ECONOMICS • The question should not be a because all the respondents respond leading question, which gives a clue from the given options. But they are about how the respondent should difficult to write as the alternatives answer. For example: should be clearly written to represent Poor Q both sides of the issue. There is also How do you like the flavour of this a possibility that the individual’s true high-quality tea? response is not present among the Good Q options given. For this, the choice of How do you like the flavour of this tea? ‘Any Other’ is provided, where the respondent can write a response, • The question should not indicate which was not anticipated by the alternatives to the answer. For researcher. Moreover, another example: Poor Q limitation of multiple-choice questions Would you like to do a job after college is that they tend to restrict the or be a housewife? answers by providing alternatives, Good Q without which the respondents may Would you like to do a job, if possible? have answered differently. The questionnaire may consist of Open-ended questions allow for closed ended (or structured) questions more individualised responses, but or open ended (or unstructured) they are difficult to interpret and hard questions. to score, since there are a lot of Closed ended or structured variations in the responses. Example, questions can either be a two-way Q. What is your view about question or a multiple choice question. globalisation? When there are only two possible answers, ‘yes’ or ‘no’, it is called a two- Mode of Data Collection way question. Have you ever come across a television When there is a possibility of more than two options of answers, multiple show in which reporters ask questions choice questions are more appropriate. from children, housewives or general Example, public regarding their examination Q. Why did you sell your land? performance or a brand of soap or a ( To pay off the debts. i) political party? The purpose of asking ii () To finance children’s educa- questions is to do a survey for tion. collection of data. There are three ii (i) To invest in another property. basic ways of collecting data: (i) i) (v Any other (please specify). Personal Interviews, (ii) Mailing Closed -ended questions are easy (questionnaire) Surveys, and (iii) to use, score and code for analysis, Telephone Interviews. COLLECTION OF DATA 1 3 Personal Interviews less expensive. It allows the researcher to have access to people in remote This method is used areas too, who might be difficult to when the researcher reach in person or by telephone. It has access to all the does not allow influencing of the members. The resea- respondents by the interviewer. It also rcher (or investigator) permits the respondents to take conducts face to face interviews with sufficient time to give thoughtful the respondents. answers to the questions. These days Personal interviews are preferred online surveys or surveys through due to various reasons. Personal short messaging service i.e. SMS have contact is made between the become popular. Do you know how an respondent and the interviewer. The online survey is conducted? interviewer has the opportunity of The disadvantages of mail survey explaining the study and answering are that, there is less opportunity to any query of the respondents. The provide assistance in clarifying interviewer can request the respon- instructions, so there is a possibility dent to expand on answers that are of misinterpretation of questions. particularly important. Misinterpre- Mailing is also likely to produce low response rates due to certain factors tation and misunderstanding can be such as returning the questionnaire avoided. Watching the reactions of the without completing it, not returning respondents can provide supplemen- the questionnaire at all, loss of tary information. questionnaire in the mail itself, etc. Personal interview has some demerits too. It is expensive, as it Telephone Interviews requires trained interviewers. It takes In a telephone interview, the longer time to complete the survey. investigator asks questions over the Presence of the researcher may inhibit telephone. The advan- respondents from saying what they tages of telephone really think. interviews are that they are cheaper than Mailing Questionnaire personal interviews and When the data in a survey are can be conducted in a shorter time. collected by mail, the questionnaire is They allow the researcher to assist the sent to each individual respondent by clarifying the by mail with a request questions. Telephone interview is to complete and return better in the cases where the it by a given date. The respondents are reluctant to answer advantages of this certain questions in personal method are that, it is interviews. 1 4 STATISTICS FOR ECONOMICS Activities small group which is known as Pilot Survey or Pre-Testing of the • You have to collect information questionnaire. The pilot survey helps from a person, who lives in a in providing a preliminary idea about remote village of India. Which the survey. It helps in pre-testing of mode of data collection will be the most appropriate for the questionnaire, so as to know the collecting information from him? shortcomings and drawbacks of the • You have to interview the parents questions. Pilot survey also helps in about the quality of teaching in assessing the suitability of questions, a school. If the principal of the clarity of instructions, performance of school is present there, what enumerators and the cost and time types of problems can arise? involved in the actual survey. The disadvantage of this method is access to people, as many people 4. CENSUS AND SAMPLE SURVEYS may not own telephones. Telephone Census or Complete Enumeration Interviews also obstruct visual A survey, which includes every reactions of the respondents, which element of the population, is known becomes helpful in obtaining as Census or the Method of Complete information on sensitive issues. Enumeration. If certain agencies are interested in studying the total Pilot Survey population in India, they have to Once the questionnaire is ready, it is obtain information from all the advisable to conduct a try-out with a households in rural and urban India. Advantages Disadvantages • Highest Response Rate • Most expensive • Allows use of all types of questions • Possibility of influencing • Better for using open-ended respondents questions • More time taking. • Allows clarification of ambiguous questions. • Least expensive • Cannot be used by illiterates • Only method to reach remote areas • Long response time • No influence on respondents • Does not allow explanation of • Maintains anonymity of respondents unambiguous questions • Best for sensitive questions. • Reactions cannot be watched. • Relatively low cost • Limited use • Relatively less influence on • Reactions cannot be watched respondents • Possibility of influencing respon- • Relatively high response rate. dents. COLLECTION OF DATA 1 5 The essential feature of this method is that this covers every individual unit in the entire population. You cannot select some and leave out others. You may be familiar with the Census of India, which is carried out every ten years. A house-to-house enquiry is carried out, covering all households in India. Demographic data on birth and death rates, literacy, workforce, life expectancy, size and composition of population, etc. are collected and Source: Census of India, 2001. published by the Registrar General of India. The last Census of India was 1981 indicated that the rate of held in February 2001. population growth during 1960s and 1970s remained almost same. 1991 Census indicated that the annual growth rate of population during 1980s was 2.14 per cent, which came down to 1.93 per cent during 1990s according to Census 2001. “At 00.00 hours of first March, 2001 the population of India stood at 1027,015,247 comprising of 531,277,078 males and 495,738,169 females. Thus, India becomes the second country in the world after China to cross the one billion mark.” Source: Census of India, 2001. Sample Survey Population or the Universe in statistics means totality of the items under According to the Census 2001, study. Thus, the Population or the population of India is 102.70 crore. It Universe is a group to which the was 23.83 crore according to Census results of the study are intended to 1901. In a period of hundred years, apply. A population is always all the the population of our country individuals/items who possess certain increased by 78.87 crore. Census characteristics (or a set of characteris- 1 6 STATISTICS FOR ECONOMICS tics), according to the purpose of the • Sample: Ten per cent of the survey. The first task in selecting a agricultural labourers in Chura- sample is to identify the population. chandpur district. Once the population is identified, the Most of the surveys are sample researcher selects a Representative surveys. These are preferred in Sample, as it is difficult to study the statistics because of a number of entire population. A sample refers to reasons. A sample can provide a group or section of the population reasonably reliable and accurate from which information is to be information at a lower cost and obtained. A good sample (represen- shorter time. As samples are smaller tative sample) is generally smaller than than population, more detailed the population and is capable of information can be collected by providing reasonably accurate conducting intensive enquiries. As we information about the population at need a smaller team of enumerators, a much lower cost and shorter time. it is easier to train them and supervise Suppose you want to study the their work more effectively. average income of people in a certain Now the question is how do you region. According to the Census do the sampling? There are two main method, you would be required to find types of sampling, random and non- out the income of every individual in random. The following description will the region, add them up and divide make their distinction clear. by number of individuals to get the average income of people in the region. Activities This method would require huge • In which years will the next expenditure, as a large number of Census be held in India and enumerators have to be employed. China? Alternatively, you select a represent- • If you have to study the opinion ative sample, of a few individuals, from of students about the new the region and find out their income. economics textbook of class XI, what will be your population and The average income of the selected sample? group of individuals is used as an • If a researcher wants to estimate estimate of average income of the the average yield of wheat in individuals of the entire region. Punjab, what will be her/his population and sample? Example • Research problem: To study the Random Sampling economic condition of agricultural As the name suggests, random labourers in Churachandpur district sampling is one where the individual of Manipur. units from the population (samples) • Population: All agricultural are selected at random. The labourers in Churachandpur district. government wants to determine the COLLECTION OF DATA 1 7 tables have been generated to guarantee equal probability of selection of every individual unit (by their listed serial number in the sampling frame) in the population. They are available either in a A Population of 20 published form or can be generated Kuchha and 20 Pucca Houses by using appropriate software packages (See Appendix B).You can start using the table from anywhere, i.e., from any page, column, row or A Representative A non Representative point. In the above example, you need Sample Sample to select a sample of 30 households impact of the rise in petrol price on out of 300 total households. Here, the the household budget of a particular largest serial number is 300, a three locality. For this, a representative digit number and therefore we consult (random) sample of 30 households has three digit random numbers in to be taken and studied. The names sequence. We will skip the random of all the 300 households of that area numbers greater than 300 since there are written on pieces of paper and is no household number greater than mixed well, then 30 names to be 300. Thus, the 30 selected households interviewed are selected one by one. are with serial numbers: 149, 219, In the random sampling, every 111, 165, 230, 007, 089, 212, 051, individual has an equal chance of being 244, 300, 051, 244, 155, 300, 051, selected and the individuals who are 152, 156, 205, 070, 015, 157, 040, selected are just like the ones who are 243, 479, 116, 122, 081, 160, 162. not selected. In the above example, all the 300 sampling units (also called sampling frame) of the population got Exit Polls an equal chance of being included in the sample of 30 units and hence the You must have seen that when an sample, such drawn, is a random election takes place, the television sample. This is also called lottery networks provide election coverage. method. The same could be done using They also try to predict the results. a Random Number Table also. This is done through exit polls, wherein a random sample of voters How to use the Random Number who exit the polling booths are asked Tables? whom they voted for. From the data of the sample of voters, the Do you know what are the Random prediction is made. Number Tables? Random number 1 8 STATISTICS FOR ECONOMICS Activity characteristic of the population (that • You have to analyse the trend of may be the average income, etc.). It is foodgrains production in India the error that occurs when you make for the last fifty years. As it is an observation from the sample taken difficult to include all the years, from the population. Thus, the you have to select a sample of difference between the actual value of production of ten years. Using a parameter of the population (which the Random Number Tables, is not known) and its estimate (from how will you select your sample? the sample) is the sampling error. It is possible to reduce the magnitude of Non-Random Sampling sampling error by taking a larger There may be a situation that you sample. have to select 10 out of 100 Example households in a locality. You have to decide which household to select and Consider a case of incomes of 5 which to reject. You may select the farmers of Manipur. The variable x households conveniently situated or (income of farmers) has measure- the households known to you or your ments 500, 550, 600, 650, 700. We friend. In this case, you are using your note that the population average of judgement (bias) in selecting 10 (500+550+600+650+700) households. This way of selecting 10 ÷ 5 = 3000 ÷ 5 = 600. out of 100 households is not a random Now, suppose we select a sample selection. In a non-random sampling of two individuals where x has method all the units of the population measurements of 500 and 600. The do not have an equal chance of being sample average is (500 + 600) ÷ 2 selected and convenience or judgement = 1100 ÷ 2 = 550. of the investigator plays an important Here, the sampling error of the role in selection of the sample. They are estimate = 600 (true value) – 550 mainly selected on the basis of (estimate) = 50. judgment, purpose, convenience or quota and are non-random samples. Non-Sampling Errors Non-sampling errors are more serious . 5 SAMPLING AND NON-S AMPLING than sampling errors because a ERRORS sampling error can be minimised by Sampling Errors taking a larger sample. It is difficult The purpose of the sample is to take to minimise non-sampling error, even an estimate of the population. by taking a large sample. Even a Sampling error refers to the Census can contain non-sampling differences between the sample errors. Some of the non-sampling estimate and the actual value of a errors are: COLLECTION OF DATA 1 9 Errors in Data Acquisition process and tabulate the statistical This type of error arises from recording data. Some of the major agencies at of incorrect responses. Suppose, the the national level are Census of India, teacher asks the students to measure National Sample Survey Organisation the length of the teacher’s table in the (NSSO), Central Statistical Organisa- classroom. The measurement by the tion (CSO), Registrar General of India students may differ. The differences (RGI), Directorate General of may occur due to differences in Commercial Intelligence and Statistics measuring tape, carelessness of the (DGCIS), Labour Bureau etc. students etc. Similarly, suppose we The Census of India provides the want to collect data on prices of most complete and continuous oranges. We know that prices vary demographic record of population. The from shop to shop and from market Census is being regularly conducted to market. Prices also vary according every ten years since 1881. The first to the quality. Therefore, we can only Census after Independence was held consider the average prices. Recording in 1951. The Census collects mistakes can also take place as the information on various aspects of enumerators or the respondents may population such as the size, density, commit errors in recording or trans- sex ratio, literacy, migration, rural- scripting the data, for example, he/ urban distribution etc. Census in she may record 13 instead of 31. India is not merely a statistical operation, the data is interpreted and Non-Response Errors analysed in an interesting manner. The NSSO was established by the Non-response occurs if an interviewer government of India to conduct is unable to contact a person listed in nation-wide surveys on socio- the sample or a person from the economic issues. The NSSO does sample refuses to respond. In this continuous surveys in successive case, the sample observation may not rounds. The data collected by NSSO be representative. surveys, on different socio economic subjects, are released through reports Sampling Bias and its quarterly journal Sampling bias occurs when the Sarvekshana. NSSO provides periodic sampling plan is such that some estimates of literacy, school members of the target population enrolment, utilisation of educational could not possibly be included in the services, employment, unemployment, sample. manufacturing and service sector enterprises, morbidity, maternity, 6. CENSUS OF INDIA AND NSSO child care, utilisation of the public There are some agencies both at the distribution system etc. The NSS 59th national and state level, which collect, round survey (January–December 2 0 STATISTICS FOR ECONOMICS 2003) was on land and livestock of data collection is to understand, holdings, debt and investment. The explain and analyse a problem and NSS 60th round survey (January– causes behind it. Primary data is June 2004) was on morbidity and obtained by conducting a survey. health care. The NSSO also Survey includes various steps, which undertakes the fieldwork of Annual need to be planned carefully. There are survey of industries, conducts crop estimation surveys, collects rural and various agencies which collect, urban retail prices for compilation of process, tabulate and publish consumer price index numbers. statistical data. These can be used as secondary data. However, the choice 7. CONCLUSION of source of data and mode of data Economic facts, expressed in terms of collection depends on the objective of numbers, are called data. The purpose the study. Recap • Data is a tool which helps in reaching a sound conclusion on any problem by providing information. • Primary data is based on first hand information. • Survey can be done by personal interviews, mailing questionnaires and telephone interviews. • Census covers every individual/unit belonging to the population. • Sample is a smaller group selected from the population from which the relevant information would be sought. • In a random sampling, every individual is given an equal chance of being selected for providing information. • Sampling error arises due to the difference between the actual population and the estimate. • Non-sampling errors can arise in data acquisition, by non-response or by bias in selection. • Census of India and National Sample Survey Organisation are two important agencies at the national level, which collect, process and tabulate data. EXERCISES . 1 Frame at least four appropriate multiple-choice options for following questions: i) ( Which of the following is the most important when you buy a new dress? COLLECTION OF DATA 2 1 i) (i How often do you use computers? ii (i) Which of the newspapers do you read regularly? i) (v Rise in the price of petrol is justified. v () What is the monthly income of your family? 2. Frame five two-way questions (with ‘Yes’ or ‘No’). i) 3. ( There are many sources of data (true/false). i) (i Telephone survey is the most suitable method of collecting data, when the population is literate and spread over a large area (true/false). ii (i) Data collected by investigator is called the secondary data (true/false). i) (v There is a certain bias involved in the non-random selection of samples (true/false). v () Non-sampling errors can be minimised by taking large samples (true/ as) fle. 4. What do you think about the following questions. Do you find any problem with these questions? If yes, how? i) ( How far do you live from the closest market? i) (i If plastic bags are only 5 percent of our garbage, should it be banned? ii (i) Wouldn’t you be opposed to increase in price of petrol? i) a (v () Do you agree with the use of chemical fertilizers? b ( ) Do you use fertilizers in your fields? c () What is the yield per hectare in your field? 5. You want to research on the popularity of Vegetable Atta Noodles among children. Design a suitable questionnaire for collecting this information. 6. In a village of 200 farms, a study was conducted to find the cropping pattern. Out of the 50 farms surveyed, 50% grew only wheat. Identify the population and the sample here. 7. Give two examples each of sample, population and variable. 8. Which of the following methods give better results and why? a () Census (b) Sample 9. Which of the following errors is more serious and why? (a) Sampling error (b) Non-Sampling error 10. Suppose there are 10 students in your class. You want to select three out of them. How many samples are possible? 11. Discuss how you would use the lottery method to select 3 students out of 10 in your class? 12. Does the lottery method always give you a random sample? Explain. 13. Explain the procedure of selecting a random sample of 3 students out of 10 in your class, by using random number tables. 14. Do samples provide better results than surveys? Give reasons for your answer. CHAPTER Organisation of Data between census and sampling. In this Studying this chapter should enable chapter, you will know how the data, you to: that you collected, are to be classified. • classify the data for further The purpose of classifying raw data is statistical analysis; • distinguish between quantitative to bring order in them so that they and qualitative classification; can be subjected to further statistical • prepare a frequency distribution analysis easily. table; Have you ever observed your local • know the technique of forming junk dealer or kabadiwallah to whom classes; you sell old newspapers, broken • be familiar with the method of tally household items, empty glass bottles, marking; plastics etc. He purchases these • differentiate between univariate things from you and sells them to and bivariate frequency distribu- tions. those who recycle them. But with so much junk in his shop it would be very difficult for him to manage his trade, . 1 INTRODUCTION if he had not organised them properly. In the previous chapter you have To ease his situation he suitably learnt about how data is collected. You groups or “classifies” various junk. also came to know the difference He puts old newspapers together and ORGANISATION OF DATA 2 3 ties them with a rope. Then collects manner. The kabadiwallah groups his all empty glass bottles in a sack. He junk in such a way that each group heaps the articles of metals in one consists of similar items. For example, corner of his shop and sorts them into under the group “Glass” he would put groups like “iron”, “copper”, empty bottles, broken mirrors and “aluminium”, “brass” etc., and so on. windowpanes etc. Similarly when you In this way he groups his junk into classify your history books under the different classes — “newspapers, group “History” you would not put a “plastics”, “glass”, “metals” etc. — and book of a different subject in that brings order in them. Once his junk group. Otherwise the entire purpose is arranged and classified, it becomes of grouping would be lost. easier for him to find a particular item Classification, therefore, is arranging that a buyer may demand. or organising similar things into groups Likewise when you arrange your or classes. schoolbooks in a certain order, it becomes easier for you to handle Activity them. You may classify them • Visit your local post-office to find out how letters are sorted. Do you know what the pin-code in a letter indicates? Ask your postman. 2. RAW DATA Like the kabadiwallah’s junk, the unclassified data or raw data are highly disorganised. They are often very large and cumbersome to handle. To draw meaningful conclusions from them is a tedious task because they according to subjects where each do not yield to statistical methods subject becomes a group or a class. easily. Therefore proper organisation So, when you need a particular book and presentation of such data is on history, for instance, all you need needed before any systematic to do is to search that book in the statistical analysis is undertaken. group “History”. Otherwise, you Hence after collecting data the next would have to search through your step is to organise and present them entire collection to find the particular in a classified form. book you are looking for. Suppose you want to know the While classification of objects or performance of students in things saves our valuable time and mathematics and you have collected effort, it is not done in an arbitrary data on marks in mathematics of 100 2 4 STATISTICS FOR ECONOMICS students of your school. If you present TABLE 3.2 them as a table, they may appear Monthly Household Expenditure (in Rupees) on Food of 50 Households something like Table 3.1. 1904 1559 3473 1735 2760 TABLE 3.1 2041 1612 1753 1855 4439 Marks in Mathematics Obtained by 100 5090 1085 1823 2346 1523 Students in an Examination 1211 1360 1110 2152 1183 1218 1315 1105 2628 2712 47 45 10 60 51 56 66 100 49 40 4248 1812 1264 1183 1171 60 59 56 55 62 48 59 55 51 41 1007 1180 1953 1137 2048 42 69 64 66 50 59 57 65 62 50 2025 1583 1324 2621 3676 64 30 37 75 17 56 20 14 55 90 1397 1832 1962 2177 2575 62 51 55 14 25 34 90 49 56 54 1293 1365 1146 3222 1396 70 47 49 82 40 82 60 85 65 66 49 44 64 69 70 48 12 28 55 65 from Table 3.1 then you have to first 49 40 25 41 71 80 0 56 14 22 arrange the marks of 100 students 66 53 46 70 43 61 59 12 30 35 45 44 57 76 82 39 32 14 90 25 either in ascending or in descending order. That is a tedious task. It Or you could have collected data becomes more tedious, if instead of on the monthly expenditure on food 100 you have the marks of a 1,000 of 50 households in your students to handle. Similarly in Table neighbourhood to know their average 3.2, you would note that it is difficult expenditure on food. The data for you to ascertain the average collected, in that case, had you monthly expenditure of 50 households. And this difficulty will go up manifold if the number was larger — say, 5,000 households. Like our kabadiwallah, who would be distressed to find a particular item when his junk becomes large and disarranged, you would face a similar situation when you try to get any information from raw data that are large. In one word, therefore, it is a tedious task to pull information from large unclassified data. The raw data are summarised, and presented as a table, would have resembled Table 3.2. Both Tables 3.1 made comprehensible by classifi- and 3.2 are raw or unclassified data. cation. When facts of similar In both the tables you find that characteristics are placed in the same numbers are not arranged in any class, it enables one to locate them order. Now if you are asked what are easily, make comparison, and draw the highest marks in mathematics inferences without any difficulty. You ORGANISATION OF DATA 2 5 have studied in Chapter 2 that the ways. Instead of classifying your books Government of India conducts Census according to subjects — “History”, of population every ten years. The raw “Geography”, “Mathematics”, “Science” data of census are so large and etc. — you could have classified them fragmented that it appears an almost author-wise in an alphabetical order. impossible task to draw any Or, you could have also classified them meaningful conclusion from them. according to the year of publication. But when the data of Census are The way you want to classify them classified according to gender, would depend on your requirement. education, marital status, occupation, Likewise the raw data could be etc., the structure and nature of classified in various ways depending population of India is, then, easily on the purpose in hand. They can be understood. grouped according to time. Such a The raw data consist of classification is known as a observations on variables. Each unit Chronological Classification. In of raw data is an observation. In Table such a classification, data are 3.1 an observation shows a particular classified either in ascending or in descending order with reference to value of the variable “marks of a time such as years, quarters, months, student in mathematics”. The raw weeks, etc. The following example data contain 100 observations on shows the population of India “marks of a student” since there are classified in terms of years. The 100 students. In Table 3.2 it shows a variable ‘population’ is a Time Series particular value of the variable as it depicts a series of values for “monthly expenditure of a household different years. on food”. The raw data in it contain 50 observations on “monthly Example 1 expenditure on food of a household” Population of India (in crores) because there are 50 households. Year Population (Crores) Activity 1951 35.7 • Collect data of total weekly 1961 43.8 expenditure of your family for a 1971 54.6 year and arrange it in a table. 1981 68.4 See how many observations you 1991 81.8 have. Arrange the data monthly 2001 102.7 and find the number of observations. In Spatial Classification the data are classified with reference to 3. CLASSIFICATION OF DATA geographical locations such as countries, states, cities, districts, etc. The groups or classes of a Example 2 shows the yield of wheat in classification can be done in various different countries. 2 6 STATISTICS FOR ECONOMICS on the basis of either the presence or the absence of a qualitative characteristic. Such a classification of data on attributes is called a Qualitative Classification. In the following example, we find population of a country is grouped on the basis of the qualitative variable “gender”. An observation could either be a male or Example 2 a female. These two characteristics Yield of Wheat for Different Countries could be further classified on the basis Country Yield of wheat (kg/acre) of marital status (a qualitative America 1925 variable) as given below: Brazil 127 China 893 Example 3 Denmark 225 France 439 Population India 862 Male Female Activities • In the time-series of Example 1, in which year do you find the Married Unmarried Married Unmarried population of India to be the minimum. Find the year when it The classification at the first stage is the maximum. is based on the presence and absence • In Example 2, find the country of an attribute i.e. male or not male whose yield of wheat is slightly (female). At the second stage, each more than that of India’s. How class — male and female, is further sub much would that be in terms of divided on the basis of the presence or percentage? absence of another attribute i.e. • Arrange the countries of whether married or unmarried. On the Example 2 in the ascending order of yield. Do the same Activity exercise for the descending order of yield. • The objects around can be grouped as either living or non- Sometimes you come across living. Is it a quantitative characteristics that cannot be classification? expressed quantitatively. Such characteristics are called Qualities or other hand, characteristics like height, Attributes. For example, nationality, weight, age, income, marks of literacy, religion, gender, marital students, etc. are quantitative in status, etc. They cannot be measured. nature. When the collected data of Yet these attributes can be classified such characteristics are grouped into ORGANISATION OF DATA 2 7 classes, the classification is a chapter, does not tell you how it varies. Quantitative Classification. Different variables vary differently and depending on the way they vary, they Example 4 are broadly classified into two types: Frequency Distribution of Marks in i ) ( Continuous and Mathematics of 100 Students ii () Discrete. Marks Frequency A continuous variable can take any 0–10 1 numerical value. It may take integral 10–20 8 20–30 6 values (1, 2, 3, 4, ...), fractional values 30–40 7 (1/2, 2/3, 3/4, ...), and values that 40–50 21 are not exact fractions ( 2 =1.414, 50–60 23 60–70 19 3 =1.732, … , 7 =2.645). For 70–80 6 example, the height of a student, as 80–90 5 he/she grows say from 90 cm to 150 90–100 4 cm, would take all the values in Total 100 between them. It can take values that are whole numbers like 90cm, 100cm, Example 4 shows quantitative 108cm, 150cm. It can also take classification of the data of marks in fractional values like 90.85 cm, 102.34 mathematics of 100 students given in cm, 149.99cm etc. that are not whole Table 3.1 as a Frequency Distribution. numbers. Thus the variable “height” is capable of Activity manifesting in • Express the values of frequency every conceivable of Example 4 as proportion or value and its percentage of total frequency. values can also Note that frequency expressed in be broken down into infinite this way is known as relative gradations. Other examples of a frequency. continuous variable are weight, time, • In Example 4, which class has distance, etc. the maximum concentration of Unlike a continuous variable, a data? Express it as percentage discrete variable can take only certain of total observations. Which class values. Its value changes only by finite has the minimum concentration “jumps”. It “jumps” from one value to of data? another but does not take any intermediate value between them. For . 4 VARIABLES: CONTINUOUS AND example, a variable like the “number DISCRETE of students in a class”, for different A simple definition of variable, classes, would assume values that are which you have read in the last only whole numbers. It cannot take 2 8 STATISTICS FOR ECONOMICS any fractional value like before we address this question, you 0.5 because “half of a must know what a frequency student” is absurd. distribution is. Therefore it cannot take a value like 25.5 between 25 5. WHAT IS A FREQUENCY DISTRIBUTION? and 26. Instead its value A frequency distribution is a could have been either 25 comprehensive way to classify raw or 26. What we observe is data of a quantitative variable. It that as its value changes shows how the different values of a from 25 to 26, the values variable (here, the marks in in between them — the fractions are mathematics scored by a student) are not taken by it. But do not have the distributed in different classes along impression that a discrete variable with their corresponding class cannot take any fractional value. frequencies. In this case we have ten Suppose X is a variable that takes classes of marks: 0–10, 10–20, … , 90– values like 1/8, 1/16, 1/32, 1/64, ... 100. The term Class Frequency means Is it a discrete variable? Yes, because the number of values in a particular though X takes fractional values it class. For example, in the class 30– cannot take any value between two 40 we find 7 values of marks from raw adjacent fractional values. It changes data in Table 3.1. They are 30, 37, 34, or “jumps” from 1/8 to 1/16 and from 30, 35, 39, 32. The frequency of the 1/16 to 1/32. But cannot take a value class: 30–40 is thus 7. But you might in between 1/8 and 1/16 or between be wondering why 40–which is 1/16 and 1/32 occurring twice in the raw data – is not included in the class 30–40. Had Activity it been included the class frequency • Distinguish the following of 30–40 would have been 9 instead variables as continuous and of 7. The puzzle would be clear to you discrete: if you are patient enough to read this Area, volume, temperature, number appearing on a dice, chapter carefully. So carry on. You will crop yield, population, rainfall, find the answer yourself. number of cars on road, age. Each class in a frequency distribution table is bounded by Class Earlier we have mentioned that Limits. Class limits are the two ends example 4 is the frequency of a class. The lowest value is called distribution of marks in mathematics the Lower Class Limit and the highest of 100 students as shown in Table 3.1. value the Upper Class Limit. For It shows how the marks of 100 example, the class limits for the class: students are grouped into classes. You 60–70 are 60 and 70. Its lower class will be wondering as to how we got it limit is 60 and its upper class limit is from the raw data of Table 3.1. But, 70. Class Interval or Class Width is ORGANISATION OF DATA 2 9 the difference between the upper class frequency distribution of the data in limit and the lower class limit. For the our example above. To obtain the class 60–70, the class interval is 10 frequency curve we plot the class (upper class limit minus lower class marks on the X-axis and frequency on ii) lmt. the Y-axis. The Class Mid-Point or Class Mark is the middle value of a class. It lies halfway between the lower class limit and the upper class limit of a class and can be ascertained in the following manner: Class Mid-Point or Class Mark = (Upper Class Limit + Lower Class ii) ..................() L m t /2. . . . . . . . . . . . . . . . . . . 1 Fig.3.1: Diagrammatic Presentation of The class mark or mid-value of Frequency Distribution of Data. each class is used to represent the How to prepare a Frequency class. Once raw data are grouped into Distribution? classes, individual observations are not used in further calculations. While preparing a frequency Instead, the class mark is used. distribution from the raw data of Table 3.1, the following four questions need TABLE 3.3 to be addressed: The Lower Class Limits, the Upper Class . 1 How many classes should we Limits and the Class Mark have? Class Frequency Lower Upper Class . 2 What should be the size of each Class Class Marks Limit Limit class? . 3 How should we determine the class 0–10 1 0 10 5 10–20 8 10 20 15 limits? 20–30 6 20 30 25 . 4 How should we get the frequency 30–40 7 30 40 35 for each class? 40–50 21 40 50 45 50–60 23 50 60 55 How many classes should we have? 60–70 19 60 70 65 70–80 6 70 80 75 Before we determine the number 80–90 5 80 90 85 of classes, we first find out as to what 90–100 4 90 100 95 extent the variable in hand changes Frequency Curve is a graphic in value. Such variations in variable’s representation of a frequency value are captured by its range. The distribution. Fig. 3.1 shows the Range is the difference between the diagrammatic presentation of the largest and the smallest values of the 3 0 STATISTICS FOR ECONOMICS variable. A large range indicates that example, suppose the range is 100 the values of the variable are widely and the class interval is 50. Then the spread. On the other hand, a small number of classes would be just 2 range indicates that the values of the (i.e.100/50 = 2). Though there is no variable are spread narrowly. In our hard-and-fast rule to determine the example the range of the variable number of classes, the rule of thumb “marks of a student” are 100 because often used is that the number of the minimum marks are 0 and the classes should be between 5 and 15. maximum marks 100. It indicates that In our example we have chosen to the variable has a large variation. have 10 classes. Since the range is 100 After obtaining the value of range, and the class interval is 10, the it becomes easier to determine the number of classes is 100/10 =10. number of classes once we decide the class interval. Note that range is the What should be the size of each sum of all class intervals. If the class class? intervals are equal then range is the product of the number of classes and The answer to this question depends class interval of a single class. on the answer to the previous question. The equality (2) shows that Range = Number of Classes × Class given the range of the variable, we can nevl ....................2 Itra ....................() determine the number of classes once we decide the class interval. Similarly, Activities we can determine the class interval Find the range of the following: once we decide the number of classes. • population of India in Example 1, Thus we find that these two decisions • yield of wheat in Example 2. are inter-linked with one another. We Given the value of range, the cannot decide on one without deciding number of classes would be large if on the other. we choose small class intervals. A In Example 4, we have the number frequency distribution with too many of classes as 10. Given the value of classes would look too large. Such a range as 100, the class intervals are distribution is not easy to handle. So we want to have a reasonably compact automatically 10 by the equality (2). set of data. On the other hand, given Note that in the present context we the value of range if we choose a class have chosen class intervals that are interval that is too large then the equal in magnitude. However we could number of classes becomes too small. have chosen class intervals that are The data set then may be too compact not of equal magnitude. In that case, and we may not like the loss of the classes would have been of information about its diversity. For unequal width. ORGANISATION OF DATA 3 1 How should we determine the class the lower class limit of that class. Had limits? we done that we would have excluded When we classify raw data of a the observation 0. The upper class continuous variable as a frequency limit of the first class: 0–10 is then distribution, we in effect, group the obtained by adding class interval with individual observations into classes. lower class limit of the class. Thus the The value of the upper class limit of a upper class limit of the first class class is obtained by adding the class becomes 0 + 10 = 10. And this proce- interval with the value of the lower dure is followed for the other classes class limit of that class. For example, as well. the upper class limit of the class 20– Have you noticed that the upper 30 is 20 + 10 = 30 where 20 is the class limit of the first class is equal to lower class limit and 10 is the class the lower class limit of the second interval. This method is repeated for class? And both are equal to 10. This other classes as well. is observed for other classes as well. But how do we decide the lower Why? The reason is that we have used class limit of the first class? That is to the Exclusive Method of classification say, why 0 is the lower class limit of of raw data. Under the method we the first class: 0–10? It is because we form classes in such a way that the chose the minimum value of the lower limit of a class coincides with variable as the lower limit of the first the upper class limit of the previous class. In fact, we could have chosen a class. value less than the minimum value of The problem, we would face next, the variable as the lower limit of the is how do we classify an observation first class. Similarly, for the upper that is not only equal to the upper class limit for the last class we could class limit of a particular class but is have chosen a value greater than the also equal to the lower class limit of maximum value of the variable. It is the next class. For example, we find important to note that, when a observation 30 to be equal to the frequency distribution is being upper class limit of the class 20–30 constructed, the class limits should and it is equal to the lower class limit be so chosen that the mid-point or class mark of each class coincide, as of class 30–40. Then, in which of the far as possible, with any value around two classes: 20–30 or 30–40 should which the data tend to be we put the observation 30? We can put concentrated. it either in class 20–30 or in class 30– In our example on marks of 100 40. It is a dilemma that one commonly students, we chose 0 as the lower limit faces while classifying data in of the first class: 0–10 because the overlapping classes. This problem is minimum marks were 0. And that is solved by the rule of classification in why, we could not have chosen 1 as the Exclusive Method. 3 2 STATISTICS FOR ECONOMICS Exclusive Method TABLE 3.4 Frequency Distribution of Incomes of 550 The classes, by this method, are Employees of a Company formed in such a way that the upper Income (Rs) Number of Employees class limit of one class equals the 800–899 50 lower class limit of the next class. In 900–999 100 this way the continuity of the data is 1000–1099 200 maintained. That is why this method 1100–1199 150 1200–1299 40 of classification is most suitable in 1300–1399 10 case of data of a continuous variable. Total 550 Under the method, the upper class limit is excluded but the lower class limit of in the class: 800–899 those employees a class is included in the interval. Thus whose income is either Rs 800, or an observation that is exactly equal between Rs 800 and Rs 899, or Rs to the upper class limit, according to 899. If the income of an employee is the method, would not be included in exactly Rs 900 then he is put in the that class but would be included in next class: 900–999. the next class. On the other hand, if it were equal to the lower class limit Adjustment in Class Interval then it would be included in that class. In our example on marks of students, A close observation of the Inclusive the observation 40, that occurs twice, Method in Table 3.4 would show that in the raw data of Table 3.1 is not though the variable “income” is a included in the class: 30–40. It is continuous variable, no such included in the next class: 40–50. That continuity is maintained when the is why we find the frequency corres- classes are made. We find “gap” or ponding to the class 30–40 to be 7 discontinuity between the upper limit instead of 9. of a class and the lower limit of the next class. For example, between the There is another method of forming upper limit of the first class: 899 and classes and it is known as the the lower limit of the second class: Inclusive Method of classification. 900, we find a “gap” of 1. Then how do we ensure the continuity of the Inclusive Method variable while classifying data? This In comparison to the exclusive method, is achieved by making an adjustment the Inclusive Method does not exclude in the class interval. The adjustment the upper class limit in a class is done in the following way: interval. It includes the upper class . 1 Find the difference between the in a class. Thus both class limits are lower limit of the second class and parts of the class interval. the upper limit of the first class. For example, in the frequency For example, in Table 3.4 the lower distribution of Table 3.4 we include limit of the second class is 900 and ORGANISATION OF DATA 3 3 the upper limit of the first class is TABLE 3.5 899. The difference between them Frequency Distribution of Incomes of 550 Employees of a Company is 1, i.e. (900 – 899 = 1) . 2 Divide the difference obtained in Income (Rs) Number of Employees (1) by two i.e. (1/2 = 0.5) 799.5–899.5 50 . 3 Subtract the value obtained in (2) 899.5–999.5 100 999.5–1099.5 200 from lower limits of all classes 1099.5–1199.5 150 (lower class limit – 0.5) 1199.5–1299.5 40 . 4 Add the value obtained in (2) to 1299.5–1399.5 10 upper limits of all classes (upper Total 550 class limit + 0.5). After the adjustment that restores continuity of data in the frequency How should we get the frequency distribution, the Table 3.4 is modified for each class? into Table 3.5 In simple terms, frequency of an After the adjustments in class observation means how many times limits, the equality (1) that determines that observation occurs in the raw the value of class-mark would be data. In our Table 3.1, we observe that modified as the following: the value 40 occurs thrice; 0 and 10 Adjusted Class Mark = (Adjusted occur only once; 49 occurs five times Upper Class Limit + Adjusted Lower and so on. Thus the frequency of 40 Class Limit)/2. is 3, 0 is 1, 10 is 1, 49 is 5 and so on. But when the data are grouped into TABLE 3.6 Tally Marking of Marks of 100 Students in Mathematics Class Observations Tally Frequency Class Mark Mark 0–10 0 / 1 5 10–20 10, 14, 17, 12, 14, 12, 14, 14 //// /// 8 15 20–30 25, 25, 20, 22, 25, 28 //// / 6 25 30–40 30, 37, 34, 39, 32, 30, 35, //// // 7 35 40–50 47, 42, 49, 49, 45, 45, 47, 44, 40, 44, //// //// //// 49, 46, 41, 40, 43, 48, 48, 49, 49, 40, //// / 41 21 45 50–60 59, 51, 53, 56, 55, 57, 55, 51, 50, 56, //// //// //// 59, 56, 59, 57, 59, 55, 56, 51, 55, 56, //// /// 55, 50, 54 23 55 60–70 60, 64, 62, 66, 69, 64, 64, 60, 66, 69, //// //// //// 62, 61, 66, 60, 65, 62, 65, 66, 65 //// 19 65 70–80 70, 75, 70, 76, 70, 71 ///// 6 75 80–90 82, 82, 82, 80, 85 //// 5 85 90–100 90, 100, 90, 90 //// 4 95 Total 100 3 4 STATISTICS FOR ECONOMICS classes as in example 3, the Class in classifying raw data though much Frequency refers to the number of is gained by summarising it as a values in a particular class. The classified data. Once the data are counting of class frequency is done by grouped into classes, an individual tally marks against the particular observation has no significance in class. further statistical calculations. In Example 4, the class 20–30 contains Finding class frequency by tally 6 observations: 25, 25, 20, 22, 25 and marking 28. So when these data are grouped A tally (/) is put against a class for as a class 20–30 in the frequency each student whose marks are distribution, the latter provides only the number of records in that class included in that class. For example, if (i.e. frequency = 6) but not their actual the marks obtained by a student are values. All values in this class are 57, we put a tally (/) against class 50 assumed to be equal to the middle –60. If the marks are 71, a tally is put value of the class interval or class against the class 70–80. If someone mark (i.e. 25). Further statistical obtains 40 marks, a tally is put calculations are based only on the against the class 40–50. Table 3.6 values of class mark and not on the shows the tally marking of marks of values of the observations in that 100 students in mathematics from class. This is true for other classes as Table 3.1. well. Thus the use of class mark The counting of tally is made easier instead of the actual values of the when four of them are put as //// observations in statistical methods and the fifth tally is placed across involves considerable loss of them as . Tallies are then counted information. as groups of five. So if there are 16 tallies in a class, we put them as Frequency distribution with / for the sake of unequal classes convenience. Thus frequency in a class is equal to the number of tallies By now you are familiar with against that class. frequency distributions of equal class intervals. You know how they are Loss of Information constructed out of raw data. But in some cases frequency distributions The classification of data as a with unequal class intervals are more frequency distribution has an appropriate. If you observe the inherent shortcoming. While it frequency distribution of Example 4, summarises the raw data making it as in Table 3.6, you will notice that concise and comprehensible, it does most of the observations are not show the details that are found in concentrated in classes 40–50, 50–60 raw data. There is a loss of information and 60–70. Their respective frequen- ORGANISATION OF DATA 3 5 cies are 21, 23 and 19. It means that terms of unequal classes. Each of the out of 100 observations, 63 classes 40–50, 50–60 and 60–70 are (21+23+19) observations are split into two classes. The class 40– concentrated in these classes. These 50 is divided into 40–45 and 45–50. classes are densely populated with The class 50–60 is divided into 50– 55 observations. Thus, 63 percent of data and 55–60. And class 60–70 is divided lie between 40 and 70. The remaining into 60–65 and 65–70. The new 37 percent of data are in classes classes 40–45, 45–50, 50–55, 55–60, 0–10, 10–20, 20–30, 30–40, 70–80, 60–65 and 65–70 have class interval 80–90 and 90–100. These classes are of 5. The other classes: 0–10, 10–20, sparsely populated with observations. 20–30, 30–40, 70–80, 80–90 and 90– Further you will also notice that 100 retain their old class interval of observations in these classes deviate 10. The last column of this table shows more from their respective class marks the new values of class marks for than in comparison to those in other these classes. Compare them with the classes. But if classes are to be formed in such a way that class marks old values of class marks in Table 3.6. coincide, as far as possible, to a value Notice that the observations in these around which the observations in a classes deviated more from their old class tend to concentrate, then in that class mark values than their new class case unequal class interval is more mark values. Thus the new class mark appropriate. values are more representative of the Table 3.7 shows the same data in these classes than the old frequency distribution of Table 3.6 in values. TABLE 3.7 Frequency Distribution of Unequal Classes Class Observations Frequency Class Mark 0–10 0 1 5 10–20 10, 14, 17, 12, 14, 12, 14, 14 8 15 20–30 25, 25, 20, 22, 25, 28 6 25 30–40 30, 37, 34, 39, 32, 30, 35, 7 35 40–45 42, 44, 40, 44, 41, 40, 43, 40, 41 9 42.5 45–50 47, 49, 49, 45, 45, 47, 49, 46, 48, 48, 49, 49 12 47.5 50–55 51, 53, 51, 50, 51, 50, 54 7 52.5 55–60 59, 56, 55, 57, 55, 56, 59, 56, 59, 57, 59, 55, 56, 55, 56, 55 16 57.5 60–65 60, 64, 62, 64, 64, 60, 62, 61, 60, 62, 10 62.5 65–70 66, 69, 66, 69, 66, 65, 65, 66, 65 9 67.5 70–80 70, 75, 70, 76, 70, 71 6 75 80–90 82, 82, 82, 80, 85 5 85 90–100 90, 100, 90, 90 4 95 Total 100 3 6 STATISTICS FOR ECONOMICS Figure 3.2 shows the frequency TABLE 3.8 curve of the distribution in Table 3.7. Frequency Array of the Size of Households The class marks of the table are Size of the Number of plotted on X-axis and the frequencies Household Households are plotted on Y-axis. 1 5 2 15 3 25 4 35 5 10 6 5 7 3 8 2 Total 100 The variable “size of the household” is a discrete variable that Fig. 3.2: Frequency Curve only takes integral values as shown in the table. Since it does not take any fractional value between two adjacent Activity integral values, there are no classes • If you compare Figure 3.2 with in this frequency array. Since there Figure 3.1, what do you observe? are no classes in a frequency array Do you find any difference between them? Can you explain there would be no class intervals. As the difference? the classes are absent in a discrete frequency distribution, there is no class mark as well. Frequency array 6. BIVARIATE FREQUENCY DISTRIBUTION So far we have discussed the classification of data for a continuous The frequency distribution of a single variable using the example of variable is called a Univariate percentage marks of 100 students in Distribution. The example 3.3 shows mathematics. For a discrete variable, the univariate distribution of the the classification of its data is known single variable “marks of a student”. as a Frequency Array. Since a discrete A Bivariate Frequency Distribution is variable takes values and not the frequency distribution of two intermediate fractional values variables. between two integral values, we have Table 3.9 shows the frequency frequencies that correspond to each distribution of two variable sales and of its integral values. advertisement expenditure (in Rs. The example in Table 3.8 lakhs) of 20 companies. The values of illustrates a Frequency Array. sales are classed in different columns ORGANISATION OF DATA 3 7 TABLE 3.9 Bivariate Frequency Distribution of Sales (in Lakh Rs) and Advertisement Expenditure (in Thousand Rs) of 20 Firms 115–125 125–135 135–145 145–155 155–165 165–175 Total 62–64 2 1 3 64–66 1 3 4 66–68 1 1 2 1 5 68–70 2 2 4 70–72 1 1 1 1 4 Total 4 5 6 3 1 1 20 and the values of advertisement unclassified. Once the data is expenditure are classed in different collected, the next step is to classify rows. Each cell shows the frequency them for further statistical analysis. of the corresponding row and column Classification brings order in the values. For example, there are 3 firms whose sales are between Rs 135–145 data. lakhs and their advertisement The chapter enables you to know how expenditures are between Rs 64–66 data can be classified through a thousands. The use of a bivariate frequency distribution in a distribution would be taken up in comprehensive manner. Once you Chapter 8 on correlation. know the techniques of classification, 7. CONCLUSION it will be easy for you to construct a The data collected from primary and frequency distribution, both for secondary sources are raw or continuous and discrete variables. Recap • Classification brings order to raw data. • A Frequency Distribution shows how the different values of a variable are distributed in different classes along with their corresponding class frequencies. • The upper class limit is excluded but lower class limit is included in the Exclusive Method. • Both the upper and the lower class limits are included in the Inclusive Method. • In a Frequency Distribution, further statistical calculations are based only on the class mark values, instead of values of the observations. • The classes should be formed in such a way that the class mark of each class comes as close as possible, to a value around which the observations in a class tend to concentrate. 3 8 STATISTICS FOR ECONOMICS EXERCISES . 1 Which of the following alternatives is true? i) ( The class midpoint is equal to: a () The average of the upper class limit and the lower class limit. b ( ) The product of upper class limit and the lower class limit. c () The ratio of the upper class limit and the lower class limit. d ( ) None of the above. i) (i The frequency distribution of two variables is known as a () Univariate Distribution b ( ) Bivariate Distribution c () Multivariate Distribution d ( ) None of the above ii (i) Statistical calculations in classified data are based on a () the actual values of observations b ( ) the upper class limits c () the lower class limits d ( ) the class midpoints i) (v Under Exclusive method, a () the upper class limit of a class is excluded in the class interval b ( ) the upper class limit of a class is included in the class interval c () the lower class limit of a class is excluded in the class interval d ( ) the lower class limit of a class is included in the class interval v () Range is the a () difference between the largest and the smallest observations b ( ) difference between the smallest and the largest observations c () average of the largest and the smallest observations d ( ) ratio of the largest to the smallest observation . 2 Can there be any advantage in classifying things? Explain with an example from your daily life. . 3 What is a variable? Distinguish between a discrete and a continuous variable. . 4 Explain the ‘exclusive’ and ‘inclusive’ methods used in classification of data. . 5 Use the data in Table 3.2 that relate to monthly household expenditure (in Rs) on food of 50 households and i) ( Obtain the range of monthly household expenditure on food. i) (i Divide the range into appropriate number of class intervals and obtain the frequency distribution of expenditure. ii (i) Find the number of households whose monthly expenditure on food is a () less than Rs 2000 b ( ) more than Rs 3000 ORGANISATION OF DATA 3 9 c () between Rs 1500 and Rs 2500 6. In a city 45 families were surveyed for the number of domestic appliances they used. Prepare a frequency array based on their replies as recorded below. 1 3 2 2 2 2 1 2 1 2 2 3 3 3 3 3 3 2 3 2 2 6 1 6 2 1 5 1 5 3 2 4 2 7 4 2 4 3 4 2 0 3 1 4 3 7. What is ‘loss of information’ in classified data? 8. Do you agree that classified data is better than raw data? 9. Distinguish between univariate and bivariate frequency distribution. 10. Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data: 28 17 15 22 29 21 23 27 18 12 7 2 9 4 6 1 8 3 10 5 20 16 12 8 4 33 27 21 15 9 3 36 27 18 9 2 4 6 32 31 29 18 14 13 15 11 9 7 1 5 37 32 28 26 24 20 19 25 19 20 Suggested Activity • From your old mark-sheets find the marks that you obtained in mathematics in the previous classes. Arrange them year-wise. Check whether the marks you have secured in the subject is a variable or not. Also see, if over the years, you have improved in mathematics. CHAPTER Presentation of Data • Textual or Descriptive presentation Studying this chapter should • Tabular presentation enable you to: • Diagrammatic presentation. • present data using tables; • represent data using appropriate diagrams. . 2 TEXTUAL PRESENTATION OF DATA In textual presentation, data are 1 INTRODUCTION . described within the text. When the quantity of data is not too large this form You have already learnt in previous of presentation is more suitable. Look chapters how data are collected and at the following cases: organised. As data are generally voluminous, they need to be put in a Case 1 compact and presentable form. This In a bandh call given on 08 September chapter deals with presentation of data 2005 protesting the hike in prices of precisely so that the voluminous data petrol and diesel, 5 petrol pumps were collected could be made usable readily found open and 17 were closed whereas and are easily comprehended. There are 2 schools were closed and remaining 9 generally three forms of presentation of schools were found open in a town of data: Bihar. PRESENTATION OF DATA 4 1 Case 2 3 rows (for male, female and total) and Census of India 2001 reported that 3 columns (for urban, rural and total). Indian population had risen to 102 It is called a 3 × 3 Table giving 9 items crore of which only 49 crore were of information in 9 boxes called the females against 53 crore males. 74 crore "cells" of the Table. Each cell gives people resided in rural India and only information that relates an attribute of 28 crore lived in towns or cities. While gender ("male", "female" or total) with a there were 62 crore non-worker number (literacy percentages of rural population against 40 crore workers in people, urban people and total). The the entire country, urban population most important advantage of tabulation had an even higher share of non- is that it organises data for further workers (19 crores) against the workers statistical treatment and decision- (9 crores) as compared to the rural making. Classification used in population where there were 31 crore tabulation is of four kinds: workers out of a 74 crore population.... • Qualitative In both the cases data have been • Quantitative presented only in the text. A serious • Temporal and drawback of this method of presentation • Spatial is that one has to go through the complete text of presentation for Qualitative classification comprehension but at the same time, it When classification is done according enables one to emphasise certain points to qualitative characteristics like social of the presentation. status, physical status, nationality, etc., it is called qualitative classification. For example, in Table 4.1 the characteris- tics for classification are sex and location which are qualitative in nature. TABLE 4.1 Literacy in Bihar by sex and location (per cent) Location Total Sex Rural Urban Male 57.70 80.80 60.32 Female 30.03 63.30 33.57 Total 44.42 72.71 47.53 . 3 TABULAR P RESENTATION OF DATA Source: Census of India 2001, Provisional In a tabular presentation, data are Population Totals. presented in rows (read horizontally) and columns (read vertically). For Quantitative classification example see Table 4.1 below tabulating In quantitative classification, the data information about literacy rates. It has are classified on the basis of 4 2 STATISTICS FOR ECONOMICS characteristics which are quantitative Temporal classification in nature. In other words these In this classification time becomes the characteristics can be measured classifying variable and data are quantitatively. For example, age, height, categorised according to time. Time production, income, etc are quantitative may be in hours, days, weeks, months, characteristics. Classes are formed by years, etc. For example, see Table 4.3. assigning limits called class limits for TABLE 4.3 the values of the characteristic under Yearly sales of a tea shop consideration. An example of from 1995 to 2000 quantitative classification is Table 4.2. Years Sale (Rs in lakhs) 1995 79.2 TABLE 4.2 1996 81.3 Distribution of 542 respondents by 1997 82.4 their age in an election study in Bihar 1998 80.5 1999 100.2 Age group No. of 2000 91.2 ys (r) respondents Per cent 20–30 3 0.55 Data Source: Unpublished data. 30–40 61 11.25 40–50 132 24.35 In this table the classifying 50–60 153 28.24 characteristic is year and takes values 60–70 140 25.83 in the scale of time. 70–80 51 9.41 80–90 2 0.37 Al l 542 100.00 Activity • Go to your library and collect Source: Assembly election Patna central data on the number of books in constituency 2005, A.N. Sinha Institute of Social Studies, Patna. economics, the library had at the end of the year for the last Here classifying characteristic is age ten years and present the data in years and is quantifiable. in a table. Activities Spatial classification • Construct a table presenting When classification is done in such a data on preferential liking of the way that place becomes the classifying students of your class for Star variable, it is called spatial News, Zee News, BBC World, classification. The place may be a CNN, Aaj Tak and DD News. village/town, block, district, state, • Prepare a table of country, etc. i ) ( heights (in cm) and Here the classifying characteristic is i) (i weights (in kg) of students country of the world. Table 4.4 is an of your class. example of spatial classification. PRESENTATION OF DATA 4 3 TABLE 4.4 ) i ( Table Number Export from India to rest of the world in one year as share of total export (per cent) Table number is assigned to a table for Destination Export share identification purpose. If more than one table is presented, it is the table USA 21.8 Germany 5.6 number that distinguishes one table Other EU 14.7 from another. It is given at the top or UK 5.7 at the beginning of the title of the table. Japan 4.9 Generally, table numbers are whole Russia 2.1 Other East Europe 0.6 numbers in ascending order if there are OPEC 10.5 many tables in a book. Subscripted Asia 19.0 numbers like 1.2, 3.1, etc. are also in Other LDCs 5.6 use for identifying the table according Others 9.5 to its location. For example, Table l Al 100.0 number 4.5 may read as fifth table (Total Exports: US $ 33658.5 million) of the fourth chapter and so on. (See Table 4.5) Activity ii () Title • Construct a table presenting The title of a table narrates about the data collected from students of contents of the table. It has to be very your class according to their clear, brief and carefully worded so that native states/residential locality. the interpretations made from the table are clear and free from any ambiguity. 4 TABULATION . OF DATA AND PARTS OF It finds place at the head of the table A TABLE succeeding the table number or just below it. (See Table 4.5). To construct a table it is important to learn first what are the parts of a good ii (i) Captions or Column Headings statistical table. When put together in At the top of each column in a table a a systematically ordered manner these column designation is given to explain parts form a table. The most simple way figures of the column. This is of conceptualising a table may be data called caption or column heading. presented in rows and columns (See Table 4.5) alongwith some explanatory notes. Tabulation can be done using one- i) (v Stubs or Row Headings way, two-way or three-way Like a caption or column heading each classification depending upon the row of the table has to be given a number of characteristics involved. A heading. The designations of the rows good table should essentially have the are also called stubs or stub items, and following: the complete left column is known as 4 4 STATISTICS FOR ECONOMICS stub column. A brief description of the were non-workers in 2001. (See Table row headings may also be given at the .) 45. left hand top in the table. (See Table 45. .) v) (i Unit of Measurement The unit of measurement of the figures v () Body of the Table in the table (actual data) should always Body of a table is the main part and it be stated alongwith the title if the unit contains the actual data. Location of does not change throughout the table. any one figure/data in the table is fixed If different units are there for rows or and determined by the row and column columns of the table, these units must of the table. For example, data in the be stated alongwith ‘stubs’ or second row and fourth column indicate ‘captions’. If figures are large, they that 25 crore females in rural India should be rounded up and the method Table Number Title ↓ ↓ Table 4.5 Population of India according to workers and non-workers by gender and location (Crore) Column Headings/Captions ↑ ↓ Units Location Gender Workers Non-worker Total Main Marginal Total Male 17 3 20 18 38 Row Headings/stubs Rural Body of the table Female 6 5 11 25 36 Total 23 8 31 43 74 → Male 7 1 8 7 15 ← Urban Female 1 0 1 12 13 Total 8 1 9 19 28 Male 24 4 28 25 53 All Female 7 5 12 37 49 Total 31 9 40 62 102 Source : Census of India 2001 ↑ Foot note : Figures are rounded to nearest crore Source note ↑ Footnote (Note : Table 4.5 presents the same data in tabular form already presented through case 2 in textual presentation of data) PRESENTATION OF DATA 4 5 of rounding should be indicated (See Diagrams may be less accurate but Table 4.5). are much more effective than tables in presenting the data. vi (i) Source Note There are various kinds of diagrams It is a brief statement or phrase in common use. Amongst them the indicating the source of data presented important ones are the following: in the table. If more than one source is ) i ( Geometric diagram there, all the sources are to be written i i () Frequency diagram in the source note. Source note is ii (i) Arithmetic line graph generally written at the bottom of the table. (See Table 4.5). Geometric Diagram Bar diagram and pie diagram come in vi i) (i Footnote the category of geometric diagram for Footnote is the last part of the table. presentation of data. The bar diagrams Footnote explains the specific feature are of three types – simple, multiple and of the data content of the table which is component bar diagrams. not self explanatory and has not been explained earlier. Bar Diagram Simple Bar Diagram Bar diagram comprises a group of Activities equispaced and equiwidth rectangular • How many rows and columns bars for each class or category of data. are essentially required to form Height or length of the bar reads the a table? magnitude of data. The lower end of the • Can the column/row headings of a table be quantitative? bar touches the base line such that the height of a bar starts from the zero unit. Bars of a bar diagram can be visually 5 DIAGRAMMATIC . PRESENTATION OF compared by their relative height and DATA accordingly data are comprehended quickly. Data for this can be of This is the third method of presenting frequency or non-frequency type. In data. This method provides the non-frequency type data a particular quickest understanding of the actual characteristic, say production, yield, situation to be explained by data in population, etc. at various points of comparison to tabular or textual time or of different states are noted and presentations. Diagrammatic presenta- corresponding bars are made of the tion of data translates quite effectively respective heights according to the the highly abstract ideas contained in values of the characteristic to construct numbers into more concrete and easily the diagram. The values of the comprehensible form. characteristics (measured or counted) 4 6 STATISTICS FOR ECONOMICS retain the identity of each value. Figure expenditure profile, export/imports 4.1 is an example of a bar diagram. over the years, etc. Activity • You had constructed a table presenting the data about the students of your class. Draw a bar diagram for the same table. Different types of data may require different modes of diagrammatical representation. Bar diagrams are suitable both for frequency type and A category that has a longer bar non-frequency type variables and (literacy of Kerala) than another attributes. Discrete variables like family category (literacy of West Bengal), has size, spots on a dice, grades in an more of the measured (or enumerated) examination, etc. and attributes such characteristics than the other. Bars as gender, religion, caste, country, etc. (also called columns) are usually used can be represented by bar diagrams. in time series data (food grain Bar diagrams are more convenient for produced between 1980–2000, non-frequency data such as income- decadal variation in work participation TABLE 4.6 Literacy Rates of Major States of India 2001 1991 Major Indian States Person Male Female Person Male Female Andhra Pradesh (AP) 60.5 70.3 50.4 44.1 55.1 32.7 Assam (AS) 63.3 71.3 54.6 52.9 61.9 43.0 Bihar (BR) 47.0 59.7 33.1 37.5 51.4 22.0 Jharkhand (JH) 53.6 67.3 38.9 41.4 55.8 31.0 Gujarat (GJ) 69.1 79.7 57.8 61.3 73.1 48.6 Haryana (HR) 67.9 78.5 55.7 55.8 69.1 40.4 Karnataka (KA) 66.6 76.1 56.9 56.0 67.3 44.3 Kerala (KE) 90.9 94.2 87.7 89.8 93.6 86.2 Madhya Pradesh (MP) 63.7 76.1 50.3 44.7 58.5 29.4 Chhattisgarh (CH) 64.7 77.4 51.9 42.9 58.1 27.5 Maharashtra (MR) 76.9 86.0 67.0 64.9 76.6 52.3 Orissa (OR) 63.1 75.3 50.5 49.1 63.1 34.7 Punjab (PB) 69.7 75.2 63.4 58.5 65.7 50.4 Rajasthan (RJ) 60.4 75.7 43.9 38.6 55.0 20.4 Tamil Nadu (TN) 73.5 82.4 64.4 62.7 73.7 51.3 Uttar Pradesh (UP) 56.3 68.8 42.2 40.7 54.8 24.4 Uttaranchal (UT) 71.6 83.3 59.6 57.8 72.9 41.7 West Bengal (WB) 68.6 77.0 59.6 57.7 67.8 46.6 India 64.8 75.3 53.7 52.2 64.1 39.3 PRESENTATION OF DATA 4 7 Fig. 4.1: Bar diagram showing literacy rates (person) of major states of India, 2001. rate, registered unemployed over the different years, marks obtained in years, literacy rates, etc.) (Fig 4.2). different subjects in different classes, Bar diagrams can have different t. ec forms such as multiple bar diagram and component bar diagram. Component Bar Diagram Activities Component bar diagrams or charts (Fig.4.3), also called sub-diagrams, are • How many states (among the very useful in comparing the sizes of major states of India) had different component parts (the elements higher female literacy rate than or parts which a thing is made up of) the national average in 2001? and also for throwing light on the • Has the gap between maximum and minimum female literacy relationship among these integral parts. rates over the states in two For example, sales proceeds from consecutive census years 2001 different products, expenditure pattern and 1991 declined? in a typical Indian family (components being food, rent, medicine, education, Multiple Bar Diagram power, etc.), budget outlay for receipts Multiple bar diagrams (Fig.4.2) are and expenditures, components of used for comparing two or more sets of labour force, population etc. data, for example income and Component bar diagrams are usually expenditure or import and export for shaded or coloured suitably. 4 8 STATISTICS FOR ECONOMICS Fig. 4.2: Multiple bar (column) diagram showing female literacy rates over two census years 1991 and 2001 by major states of India. Interpretation: It can be very easily derived from Figure 4.2 that female literacy rate over the years was on increase throughout the country. Similar other interpretations can be made from the figure like the state of Rajasthan experienced the sharpest rise in female literacy, etc. TABLE 4.7 its height equivalent to the total value Enrolment by gender at schools (per cent) of the bar [for per cent data the bar of children aged 6–14 years in a district of Bihar height is of 100 units (Figure 4.3)]. Otherwise the height is equated to total Enrolled Out of school Gender (per cent) (per cent) value of the bar and proportional heights of the components are worked Boy 91.5 8.5 out using unitary method. Smaller il Gr 58.6 41.4 Al l 78.0 22.0 components are given priority in parting the bar. Data Source: Unpublished data Pie Diagram A component bar diagram shows the bar and its sub-divisions into two A pie diagram is also a component or more components. For example, the bar might show the total population of children in the age-group of 6–14 years. The components show the proportion of those who are enrolled and those who are not. A component bar diagram might also contain different component bars for boys, girls and the total of children in the given age group range, as shown in Figure 4.3. To construct a component bar diagram, first of all, a Fig. 4.3: Enrolment at primary level in a district bar is constructed on the x-axis with of Bihar (Component Bar Diagram) PRESENTATION OF DATA 4 9 diagram, but unlike a component bar of the components have to be converted diagram, a circle whose area is into percentages before they can be proportionally divided among the used for a pie diagram. components (Fig.4.4) it represents. It TABLE 4.8 Distribution of Indian population by their working status (crore) Status Population Per cent Angular Component Marginal Worker 9 8.8 32° Main Worker 31 30.4 109° Non-Worker 62 60.8 219° l Al 102 100.0 360° is also called a pie chart. The circle is divided into as many parts as there are components by drawing straight lines from the centre to the circumference. Pie charts usually are not drawn with absolute values of a category. The values of each category are first Fig. 4.4: Pie diagram for different categories of Indian population according to working status expressed as percentage of the total 2001. value of all the categories. A circle in a pie chart, irrespective of its value of Activities radius, is thought of having 100 equal parts of 3.6° (360°/100) each. To find • Represent data presented out the angle, the component shall through Figure 4.4 by a component bar diagram. subtend at the centre of the circle, each • Does the area of a pie have any percentage figure of every component bearing on total value of the is multiplied by 3.6°. An example of this data to be represented by the conversion of percentages of pie diagram? components into angular components of the circle is shown in Table 4.8. Frequency Diagram It may be interesting to note that Data in the form of grouped frequency data represented by a component bar distributions are generally represented diagram can also be represented by frequency diagrams like histogram, equally well by a pie chart, the only frequency polygon, frequency curve requirement being that absolute values and ogive. 5 0 STATISTICS FOR ECONOMICS Histogram TABLE 4.9 Distribution of daily wage earners in a A histogram is a two dimensional locality of a town diagram. It is a set of rectangles with Daily No. Cumulative Frequencey bases as the intervals between class earning of wage 'Less than' 'More than' R) (s earners (f) boundaries (along X-axis) and with 45–49 2 2 85 areas proportional to the class 50–54 3 5 83 frequency (Fig.4.5). If the class intervals 55–59 5 10 80 are of equal width, which they generally 60–64 3 13 75 65–69 6 19 72 are, the area of the rectangles are 70–74 7 26 66 proportional to their respective 75–79 12 38 59 frequencies. However, in some type of 80–84 13 51 47 85–89 9 60 34 data, it is convenient, at times 90–94 7 67 25 necessary, to use varying width of class 95–99 6 73 18 intervals. For example, when tabulating 100–104 4 77 12 105–109 2 79 8 deaths by age at death, it would be very 110–114 3 82 6 meaningful as well as useful too to have 115–119 3 85 3 very short age intervals (0, 1, 2, ..., yrs/ Source: Unpublished data 0, 7, 28, ..., days) at the beginning when death rates are very high Since histograms are rectangles, a line parallel to the base line and of the same compared to deaths at most other magnitude is to be drawn at a vertical higher age segments of the population. distance equal to frequency (or For graphical representation of such frequency density) of the class interval. data, height for area of a rectangle is A histogram is never drawn for a the quotient of height (here frequency) discrete variable/data. Since in an and base (here width of the class interval or ratio scale the lower class interval). When intervals are equal, that boundary of a class interval fuses with the upper class boundary of the is, when all rectangles have the same previous interval, equal or unequal, the base, area can conveniently be rectangles are all adjacent and there is represented by the frequency of any no open space between two consecutive interval for purposes of comparison. rectangles. If the classes are not When bases vary in their width, the continuous they are first converted into heights of rectangles are to be adjusted continuous classes as discussed in to yield comparable measurements. Chapter 3. Sometimes the common portion between two adjacent The answer in such a situation is rectangles (Fig.4.6) is omitted giving a frequency density (class frequency better impression of continuity. The divided by width of the class interval) resulting figure gives the impression of instead of absolute frequency. a double staircase. PRESENTATION OF DATA 5 1 A histogram looks similar to a bar continuous variables, but histogram is diagram. But there are more differences drawn only for a continuous variable. than similarities between the two than Histogram also gives value of mode of it may appear at the first impression. the frequency distribution graphically The spacing and the width or the area as shown in Figure 4.5 and the x- of bars are all arbitrary. It is the height coordinate of the dotted vertical line and not the width or the area of the bar gives the mode. that really matters. A single vertical line could have served the same purpose Frequency Polygon as a bar of same width. Moreover, in A frequency polygon is a plane histogram no space is left in between bounded by straight lines, usually four two rectangles, but in a bar diagram or more lines. Frequency polygon is an some space must be left between alternative to histogram and is also consecutive bars (except in multiple derived from histogram itself. A bar or component bar diagram). frequency polygon can be fitted to a Although the bars have the same histogram for studying the shape of the width, the width of a bar is unimportant curve. The simplest method of drawing for the purpose of comparison. The a frequency polygon is to join the width in a histogram is as important midpoints of the topside of the as its height. We can have a bar consecutive rectangles of the diagram both for discrete and histogram. It leaves us with the two Fig. 4.5: Histogram for the distribution of 85 daily wage earners in a locality of a town. 5 2 STATISTICS FOR ECONOMICS ends away from the base line, denying No matter whether class boundaries or the calculation of the area under the midpoints are used in the X-axis, curve. The solution is to join the two frequencies (as ordinates) are always end-points thus obtained to the base plotted against the mid-point of class line at the mid-values of the two classes intervals. When all the points have been with zero frequency immediately at plotted in the graph, they are carefully each end of the distribution. Broken joined by a series of short straight lines. lines or dots may join the two ends with Broken lines join midpoints of two the base line. Now the total area under intervals, one in the beginning and the the curve, like the area in the other at the end, with the two ends of histogram, represents the total the plotted curve (Fig.4.6). When frequency or sample size. comparing two or more distributions Frequency polygon is the most plotted on the same axes, frequency common method of presenting grouped polygon is likely to be more useful since frequency distribution. Both class the vertical and horizontal lines of two boundaries and class-marks can be or more distributions may coincide in used along the X-axis, the distances a histogram. between two consecutive class marks Frequency Curve being proportional/equal to the width of the class intervals. Plotting of data The frequency curve is obtained by becomes easier if the class-marks fall drawing a smooth freehand curve on the heavy lines of the graph paper. passing through the points of the Fig. 4.6: Frequency polygon drawn for the data given in Table 4.9 PRESENTATION OF DATA 5 3 Fig. 4.7: Frequency curve for Table 4.9 frequency polygon as closely as frequencies are plotted against the possible. It may not necessarily pass respective lower limits of the class through all the points of the frequency interval. An interesting feature of the polygon but it passes through them as two ogives together is that their closely as possible (Fig. 4.7). intersection point gives the median Fig. 4.8 (b) of the frequency distribu- Ogive tion. As the shapes of the two ogives Ogive is also called cumulative suggest, less than ogive is never frequency curve. As there are two types decreasing and more than ogive is of cumulative frequencies, for example never increasing. less than type and more than type, TABLE 4.10 accordingly there are two ogives for any Frequency distribution of marks grouped frequency distribution data. obtained in mathematics Here in place of simple frequencies as Marks Number of ‘Less than’ ‘More than’ in the case of frequency polygon, students cumulative cumulative cumulative frequencies are plotted x f frequency frequency along y-axis against class limits of the 0–20 6 6 64 frequency distribution. For less than 20–40 5 11 58 40–60 33 44 53 ogive the cumulative frequencies are 60–80 14 58 20 plotted against the respective upper 80–100 6 64 6 limits of the class intervals whereas for Total 64 more than ogives the cumulative 5 4 STATISTICS FOR ECONOMICS Fig. 4.8(a): 'Less than' and 'More than' ogive for data given in Table 4.10 Arithmetic Line Graph An arithmetic line graph is also called time series graph and is a method of diagrammatic presentation of data. In it, time (hour, day/date, week, month, year, etc.) is plotted along x-axis and the value of the variable (time series data) along y-axis. A line graph by joining these plotted points, thus, obtained is called arithmetic line graph (time series graph). It helps in understanding the trend, periodicity, etc. in a long term time series data. Activity • Can the ogive be helpful in locating the partition values of Fig. 4.8(b): ‘Less than’ and ‘More than’ ogive the distribution it represents? for data given in Table 4.10 PRESENTATION OF DATA 5 5 TABLE 4.11 Here you can see from Fig. 4.9 that Value of Exports and Imports of India for the period 1978 to 1999, although (Rs in 100 crores) the imports were more than the exports Year Exports Imports all through, the rate of acceleration 1977–78 54 60 went on increasing after 1988–89 and 1978–79 57 68 1979–80 64 91 the gap between the two (imports and 1980–81 67 125 exports) was widened after 1995. 1982–83 88 143 1983–84 98 158 . 6 CONCLUSION 1984–85 117 171 1985–86 109 197 By now you must have been able to 1986–87 125 201 learn how collected data could be 1987–88 157 222 1988–89 202 282 presented using various forms of 1989–90 277 353 presentation — textual, tabular and 1990–91 326 432 diagrammatic. You are now also able 1991–92 440 479 to make an appropriate choice of the 1992–93 532 634 1993–94 698 731 form of data presentation as well as the 1994–95 827 900 type of diagram to be used for a given 1995–96 1064 1227 set of data. Thus you can make 1996–97 1186 1369 1997–98 1301 1542 presentation of data meaningful, 1998–99 1416 1761 comprehensive and purposeful. Scale: 1cm=200 crores on Y-axis 2000 1800 1600 1400 Values (in Rs 100 Crores) 1200 Exports Imports 1000 800 600 400 200 0 1981 1978 1979 1980 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Year Fig. 4.9: Arithmetic line graph for time series data given in Table 4.11 5 6 STATISTICS FOR ECONOMICS Recap • Data (even voluminous data) speak meaningfully through presentation. • For small data (quantity) textual presentation serves the purpose better. • For large quantity of data tabular presentation helps in accommodating any volume of data for one or more variables. • Tabulated data can be presented through diagrams which enable quicker comprehension of the facts presented otherwise. EXERCISES Answer the following questions, 1 to 10, choosing the correct answer . 1 Bar diagram is a i) ( one-dimensional diagram i) (i two-dimensional diagram ii (i) diagram with no dimension i) (v none of the above . 2 Data represented through a histogram can help in finding graphically the i) ( mean i) (i mode ii (i) median i) (v all the above . 3 Ogives can be helpful in locating graphically the i) ( mode i) (i m e a n ii (i) median i) (v none of the above . 4 Data represented through arithmetic line graph help in understanding i) ( long term trend i) (i cyclicity in data ii (i) seasonality in data i) (v all the above . 5 Width of bars in a bar diagram need not be equal (True/False). . 6 Width of rectangles in a histogram should essentially be equal (True/ False). . 7 Histogram can only be formed with continuous classification of data (True/False). PRESENTATION OF DATA 5 7 8. Histogram and column diagram are the same method of presentation of data. (True/False). 9. Mode of a frequency distribution can be known graphically with the help of histogram. (True/False). 10. Median of a frequency distribution cannot be known from the ogives. (True/False). 11. What kind of diagrams are more effective in representing the following? i) ( Monthly rainfall in a year i) (i Composition of the population of Delhi by religion ii (i) Components of cost in a factory 12. Suppose you want to emphasise the increase in the share of urban non-workers and lower level of urbanisation in India as shown in Example 4.2. How would you do it in the tabular form? 13. How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table? 14. The Indian Sugar Mills Association reported that, ‘Sugar production during the first fortnight of December 2001 was about 3,87,000 tonnes, as against 3,78,000 tonnes during the same fortnight last year (2000). The off-take of sugar from factories during the first fortnight of December 2001 was 2,83,000 tonnes for internal consumption and 41,000 tonnes for exports as against 1,54,000 tonnes for internal consumption and nil for exports during the same fortnight last season.’ i) ( Present the data in tabular form. i) (i Suppose you were to present these data in diagrammatic form which of the diagrams would you use and why? ii (i) Present these data diagrammatically. 15. The following table shows the estimated sectoral real growth rates (percentage change over the previous year) in GDP at factor cost. Year Agriculture and allied sectors Industry Services 1 () 2 () 3 () 4 () 1994–95 5.0 9.2 7.0 1995–96 –0.9 11.8 10.3 1996–97 9.6 6.0 7.1 1997–98 –1.9 5.9 9.0 1998–99 7.2 4.0 8.3 1999–2000 0.8 6.9 8.2 Represent the data as multiple time series graphs. CHAPTER Measures of Central Tendency Studying this chapter should of the data. In this chapter, you will enable you to: study the measures of central • understand the need for tendency which is a numerical method summarising a set of data by one to explain the data in brief. You can single number; see examples of summarising a large • recognise and distinguish set of data in day to day life like between the different types of average marks obtained by students averages; of a class in a test, average rainfall in • learn to compute different types of averages; an area, average production in a • draw meaningful conclusions factory, average income of persons from a set of data; living in a locality or working in a firm • develop an understanding of t. ec which type of average would be Baiju is a farmer. He grows food most useful in a particular grains in his land in a village called situation. Balapur in Buxar district of Bihar. The village consists of 50 small farmers. Baiju has 1 acre of land. You are . 1 INTRODUCTION interested in knowing the economic In the previous chapter, you have read condition of small farmers of Balapur. the tabular and graphic representation You want to compare the economic MEASURES OF CENTRAL TENDENCY 5 9 condition of Baiju in Balapur village. 2. ARITHMETIC MEAN For this, you may have to evaluate the Suppose the monthly income (in Rs) size of his land holding, by comparing of six families is given as: with the size of land holdings of other 1600, 1500, 1400, 1525, 1625, 1630. farmers of Balapur. You may like to The mean family income is see if the land owned by Baiju is – 1 above average in ordinary sense . obtained by adding up the incomes (see the Arithmetic Mean below) and dividing by the number of 2 above the size of what half the . families. farmers own (see the Median 1600 + 1500 + 1400 + 1525 + 1625 + 1630 Rs below) 6 3 above what most of the farmers . = Rs 1,547 own (see the Mode below) It implies that on an average, a In order to evaluate Baiju’s relative family earns Rs 1,547. economic condition, you will have to Arithmetic mean is the most summarise the whole set of data of commonly used measure of central land holdings of the farmers of tendency. It is defined as the sum of Balapur. This can be done by use of the values of all observations divided central tendency, which summarises by the number of observations and is the data in a single value in such a usually denoted by x . In general, if way that this single value can there are N observations as X1, X2, X3, represent the entire data. The ..., XN, then the Arithmetic Mean is measuring of central tendency is a given by way of summarising the data in the form of a typical or representative X 1 + X 2 + X 3 + ... + X N x= value. N There are several statistical SX = measures of central tendency or N “averages”. The three most commonly Where, S X = sum of all observa- used averages are: tions and N = total number of obser- • Arithmetic Mean vations. • Median • Mode How Arithmetic Mean is Calculated You should note that there are two more types of averages i.e. Geometric The calculation of arithmetic mean Mean and Harmonic Mean, which are can be studied under two broad suitable in certain situations. categories: However, the present discussion will . 1 Arithmetic Mean for Ungrouped be limited to the three types of Data. averages mentioned above. . 2 Arithmetic Mean for Grouped Data. 6 0 STATISTICS FOR ECONOMICS Arithmetic Mean for Series of mean by direct method. The Ungrouped Data computation can be made easier by using assumed mean method. Direct Method In order to save time of calculation of mean from a data set containing a Arithmetic mean by direct method is large number of observations as well the sum of all observations in a series as large numerical figures, you can divided by the total number of use assumed mean method. Here you observations. assume a particular figure in the data as the arithmetic mean on the basis Example 1 of logic/experience. Then you may Calculate Arithmetic Mean from the take deviations of the said assumed data showing marks of students in a mean from each of the observation. class in an economics test: 40, 50, 55, You can, then, take the summation of 78, 58. these deviations and divide it by the number of observations in the data. SX X= The actual arithmetic mean is N estimated by taking the sum of the 40 + 50 + 55 + 78 + 58 assumed mean and the ratio of sum = = 56.2 of deviations to number of observa- 5 tions. Symbolically, The average marks of students in Let, A = assumed mean the economics test are 56.2. X = individual observations N = total numbers of observa- Assumed Mean Method tions If the number of observations in the d = deviation of assumed mean data is more and/or figures are large, from individual observation, it is difficult to compute arithmetic i.e. d = X – A (HEIGHT IN INCHES) MEASURES OF CENTRAL TENDENCY 6 1 Then sum of all deviations is taken Arithmetic Mean using assumed mean as Sd = S( X - A ) method Sd Sd X =A + = 850 + (2, 660)/10 Then find N N Sd = Rs1,116. Then add A and to get X N Thus, the average weekly income Sd of a family by both methods is Therefore, X = A + Rs 1,116. You can check this by using N You should remember that any the direct method. value, whether existing in the data or not, can be taken as assumed mean. Step Deviation Method However, in order to simplify the The calculations can be further calculation, centrally located value in simplified by dividing all the deviations the data can be selected as assumed taken from assumed mean by the mean. common factor ‘c’. The objective is to Example 2 avoid large numerical figures, i.e., if d = X – A is very large, then find d'. The following data shows the weekly This can be done as follows: income of 10 families. Family d X-A A B C D E F G H = . c C I J Weekly Income (in Rs) The formula is given below: 850 700 100 750 5000 80 420 2500 S d¢ 400 360 X =A + ·c Compute mean family income. N Where d' = (X – A)/c, c = common TABLE 5.1 Computation of Arithmetic Mean by factor, N = number of observations, Assumed Mean Method A= Assumed mean. Families Income d = X – 850 d ' Thus, you can calculate the () X = (X – 850)/10 arithmetic mean in the example 2, by A 850 0 0 the step deviation method, B 700 –150 –15 X = 850 + (266)/10 × 10 = Rs 1,116. C 100 –750 –75 D 750 –100 –10 Calculation of arithmetic mean for E 5000 +4150 +415 Grouped data F 80 –770 –77 G 420 –430 –43 Discrete Series H 2500 +1650 +165 I 400 –450 –45 Direct Method J 360 –490 –49 In case of discrete series, frequency 11160 +2660 +266 against each of the observations is 6 2 STATISTICS FOR ECONOMICS multiplied by the value of the Assumed Mean Method observation. The values, so obtained, As in case of individual series the are summed up and divided by the calculations can be simplified by using total number of frequencies. assumed mean method, as described Symbolically, earlier, with a simple modification. Since frequency (f) of each item is S fX given here, we multiply each deviation X = Sf (d) by the frequency to get fd. Then we Where, S fX = sum of product of get S fd. The next step is to get the variables and frequencies. total of all frequencies i.e. S f. Then S f = sum of frequencies. find out S fd/ S f. Finally the arithmetic mean is calculated by Example 3 S fd Calculate mean farm size of X =A + using assumed mean cultivating households in a village for Sf the following data. method. Farm Size (in acres): 64 63 62 61 60 59 Step Deviation Method No. of Cultivating Households: In this case the deviations are divided 8 18 12 9 7 6 by the common factor ‘c’ which simplifies the calculation. Here we TABLE 5.2 d X-A Computation of Arithmetic Mean by estimate d' = = in order to Direct Method c C Farm Size No. of X d fd reduce the size of numerical figures () X cultivating (1 × 2) (X - 62) (2 × 4) for easier calculation. Then get fd' and in acres households(f) () 1 () 2 () 3 () 4 5 () S fd'. Finally the formula for step 64 8 512 +2 +16 deviation method is given as, 63 18 1134 +1 +18 S fd ¢ 62 12 744 0 0 X =A + ·c 61 9 549 –1 –9 Sf 60 7 420 –2 –14 59 6 354 –3 –18 Activity 60 3713 –3 –7 • Find the mean farm size for the data given in example 3, by using Arithmetic mean using direct method, step deviation and assumed mean methods. S fX 3717 X = = = 61.88 acres Sf 60 Continuous Series Therefore, the mean farm size in a Here, class intervals are given. The village is 61.88 acres. process of calculating arithmetic mean MEASURES OF CENTRAL TENDENCY 6 3 in case of continuous series is same Steps: as that of a discrete series. The only . 1 Obtain mid values for each class difference is that the mid-points of denoted by m. various class intervals are taken. You 2 Obtain S fm and apply the direct . should note that class intervals may method formula: be exclusive or inclusive or of unequal size. Example of exclusive class S fm 2110 X= = = 30.14 marks interval is, say, 0–10, 10–20 and so Sf 70 on. Example of inclusive class interval is, say, 0–9, 10–19 and so on. Example Step deviation method of unequal class interval is, say, 0–20, 20–50 and so on. In all these m A . 1 Obtain d' = cases, calculation of arithmetic mean c is done in a similar way. . 2 Take A = 35, (any arbitrary figure), c = common factor. Example 4 £ fd’ ( 34) Calculate average marks of the X = A+ c = 35 + 10 £f 70 following students using (a) Direct = 30.14 marks method (b) Step deviation method. An interesting property of A.M. Direct Method Marks It is interesting to know and 0–10 10–20 20–30 30–40 40–50 useful for checking your calculation 50–60 60–70 that the sum of deviations of items No. of Students about arithmetic mean is always equal 5 12 15 25 8 3 2 to zero. Symbolically, S ( X – X ) = 0. However, arithmetic mean is TABLE 5.3 affected by extreme values. Any large Computation of Average Marks for Exclusive Class Interval by Direct Method value, on either end, can push it up or down. Mark No. of mid d fm d'=(m-35) f' x () students value (2)×(3) 10 ( f) (m) Weighted Arithmetic Mean 1 () 2 () 3 () () 4 5 () 6 () Sometimes it is important to assign 0–10 5 5 25 –3 –15 10–20 12 15 180 –2 –24 weights to various items according to 20–30 15 25 375 –1 –15 their importance, when you calculate 30–40 25 35 875 0 0 the arithmetic mean. For example, 40–50 8 45 360 1 8 there are two commodities, mangoes 50–60 3 55 165 2 6 60–70 2 65 130 3 6 and potatoes. You are interested in finding the average price of mangoes 70 2110 –34 (p1) and potatoes (p2). The arithmetic 6 4 STATISTICS FOR ECONOMICS p1 + p2 3. MEDIAN mean will be . However, you 2 The arithmetic mean is affected by the might want to give more importance presence of extreme values in the data. to the rise in price of potatoes (p2). To If you take a measure of central do this, you may use as ‘weights’ the tendency which is based on middle quantity of mangoes (q1) and the position of the data, it is not affected quantity of potatoes (q2). Now the by extreme items. Median is that arithmetic mean weighted by the positional value of the variable which divides the distribution into two equal q1p1 + q 2 p 2 quantities would be . parts, one part comprises all values q1 + q 2 greater than or equal to the median In general the weighted arithmetic value and the other comprises all mean is given by, values less than or equal to it. The w1 x1 + w 2 x 2 +...+ w n x n £ wx Median is the “middle” element when = the data set is arranged in order of the w1 + w 2 +...+ w n £w magnitude. When the prices rise, you may be interested in the rise in the price of Computation of median the commodities that are more The median can be easily computed important to you. You will read more by sorting the data from smallest to about it in the discussion of Index largest and counting the middle value. Numbers in Chapter 8. Example 5 Activities Suppose we have the following • Check this property of the observation in a data set: 5, 7, 6, 1, 8, arithmetic mean for the following 10, 12, 4, and 3. example: Arranging the data, in ascending order X: 4 6 8 10 12 you have: • In the above example if mean is 1, 3, 4, 5, 6, 7, 8, 10, 12. increased by 2, then what happens to the individual observations, if all are equally affected. The “middle score” is 6, so the • If first three items increase by median is 6. Half of the scores are 2, then what should be the larger than 6 and half of the scores values of the last two items, so are smaller. that mean remains the same. If there are even numbers in the • Replace the value 12 by 96. What data, there will be two observations happens to the arithmetic mean. which fall in the middle. The median Comment. in this case is computed as the MEASURES OF CENTRAL TENDENCY 6 5 arithmetic mean of the two middle th (N+1) values. Median = size of item 2 Example 6 Discrete Series The following data provides marks of In case of discrete series the position 20 students. You are required to of median i.e. (N+1)/2th item can be calculate the median marks. located through cumulative freque- 25, 72, 28, 65, 29, 60, 30, 54, 32, 53, ncy. The corresponding value at this 33, 52, 35, 51, 42, 48, 45, 47, 46, 33. position is the value of median. Arranging the data in an ascending Example 7 order, you get The frequency distribution of the 25, 28, 29, 30, 32, 33, 33, 35, 42, number of persons and their 45, 46, 47, 48, 51, 52, 53, 54, 60, respective incomes (in Rs) are given below. Calculate the median income. Income (in Rs): 10 20 30 40 65, 72. Number of persons: 2 4 10 4 You can see that there are two In order to calculate the median observations in the middle, namely 45 income, you may prepare the and 46. The median can be obtained frequency distribution as given below. by taking the mean of the two observations: TABLE 5.4 Computation of Median for Discrete Series 45 + 46 Median = = 45.5 marks Income No of Cumulative 2 (in Rs) persons(f) frequency(cf) In order to calculate median it is 10 2 2 important to know the position of the 20 4 6 median i.e. item/items at which the 30 10 16 40 4 20 median lies. The position of the median can be calculated by the The median is located in the (N+1)/ following formula: 2 = (20+1)/2 = 10.5th observation. th This can be easily located through (N+1) cumulative frequency. The 10.5th Position of median = item 2 observation lies in the c.f. of 16. The Where N = number of items. income corresponding to this is Rs 30, You may note that the above so the median income is Rs 30. formula gives you the position of the median in an ordered array, not the Continuous Series median itself. Median is computed by In case of continuous series you have the formula: to locate the median class where 6 6 STATISTICS FOR ECONOMICS N/2th item [not (N+1)/2th item] lies. In the above illustration median The median can then be obtained as class is the value of (N/2)th item follows: (i.e.160/2) = 80th item of the series, (N/2 c.f.) which lies in 35–40 class interval. Median = L + h Applying the formula of the median f Where, L = lower limit of the median as: class, TABLE 5.5 c.f. = cumulative frequency of the class Computation of Median for Continuous preceding the median class, Series f = frequency of the median class, Daily wages No. of Cumulative h = magnitude of the median class (in Rs) Workers (f) Frequency interval. 20–25 14 14 No adjustment is required if 25–30 28 42 frequency is of unequal size or 30–35 33 75 magnitude. 35–40 30 105 40–45 20 125 Example 8 45–50 15 140 50–55 13 153 Following data relates to daily wages 55–60 7 160 of persons working in a factory. Compute the median daily wage. (N/2 c.f.) Median = L + h Daily wages (in Rs): f 55–60 50–55 45–50 40–45 35–40 30–35 35 +(80 75) 25–30 20–25 = (40 35) Number of workers: 30 7 13 15 20 30 33 = Rs 35.83 28 14 Thus, the median daily wage is The data is arranged in ascending order here. Rs 35.83. This means that 50% of the MEASURES OF CENTRAL TENDENCY 6 7 workers are getting less than or equal The third Quartile (denoted by Q3) or to Rs 35.83 and 50% of the workers upper Quartile has 75% of the items are getting more than or equal to this of the distribution below it and 25% wage. of the items above it. Thus, Q1 and Q3 You should remember that denote the two limits within which median, as a measure of central central 50% of the data lies. tendency, is not sensitive to all the values in the series. It concentrates on the values of the central items of the data. Activities • Find mean and median for all four values of the series. What do you observe? TABLE 5.6 Percentiles Mean and Median of different series Percentiles divide the distribution into Series X (Variable Mean Median hundred equal parts, so you can get Values) 99 dividing positions denoted by P1, A 1, 2, 3 ? ? P2, P3, ..., P99. P50 is the median value. B 1, 2, 30 ? ? C 1, 2, 300 ? ? If you have secured 82 percentile in a D 1, 2, 3000 ? ? management entrance examination, it means that your position is below 18 • Is median affected by extreme values? What are outliers? percent of total candidates appeared • Is median a better method than in the examination. If a total of one mean? lakh students appeared, where do you stand? Quartiles Calculation of Quartiles Quartiles are the measures which divide the data into four equal parts, The method for locating the Quartile each portion contains equal number is same as that of the median in case of observations. Thus, there are three of individual and discrete series. The quartiles. The first Quartile (denoted value of Q1 and Q3 of an ordered series by Q1) or lower quartile has 25% of can be obtained by the following the items of the distribution below it formula where N is the number of and 75% of the items are greater than observations. it. The second Quartile (denoted by Q2) or median has 50% of items below it (N + 1)th and 50% of the observations above it. Q1= size of item 4 6 8 STATISTICS FOR ECONOMICS 3(N +1)th Mode is the most frequently observed Q3 = size of item. data value. It is denoted by Mo. 4 Computation of Mode Example 9 Discrete Series Calculate the value of lower quartile from the data of the marks obtained Consider the data set 1, 2, 3, 4, 4, 5. by ten students in an examination. The mode for this data is 4 because 4 22, 26, 14, 30, 18, 11, 35, 41, 12, 32. occurs most frequently (twice) in the Arranging the data in an ascending data. order, 11, 12, 14, 18, 22, 26, 30, 32, 35, 41. Example 10 (N +1)th Look at the following discrete series: Q1 = size of item = size of 4 Variable 10 20 30 40 50 Frequency 2 8 20 10 5 (10 +1)th item = size of 2.75th item Here, as you can see the maximum 4 frequency is 20, the value of mode is = 2nd item + .75 (3rd item – 2nd item) 30. In this case, as there is a unique = 12 + .75(14 –12) = 13.5 marks. value of mode, the data is unimodal. But, the mode is not necessarily Activity unique, unlike arithmetic mean and • Find out Q3 yourself. median. You can have data with two modes (bi-modal) or more than two 5. MODE modes (multi-modal). It may be possible that there may be no mode if Sometimes, you may be interested in no value appears more frequent than knowing the most typical value of a any other value in the distribution. For series or the value around which example, in a series 1, 1, 2, 2, 3, 3, 4, maximum concentration of items 4, there is no mode. occurs. For example, a manufacturer would like to know the size of shoes that has maximum demand or style of the shirt that is more frequently demanded. Here, Mode is the most Unimodal Data Bimodal Data appropriate measure. The word mode has been derived from the French Continuous Series word “la Mode” which signifies the In case of continuous frequency most fashionable values of a distribution, modal class is the class distribution, because it is repeated the with largest frequency. Mode can be highest number of times in the series. calculated by using the formula: MEASURES OF CENTRAL TENDENCY 6 9 exclusive to calculate the mode. If mid D1 MO = L + h points are given, class intervals are D1 + D2 to be obtained. Where L = lower limit of the modal class Example 11 D 1 = difference between the frequency Calculate the value of modal worker of the modal class and the frequency family’s monthly income from the of the class preceding the modal class following data: (ignoring signs). Income per month (in ’000 Rs) D2 = difference between the frequency Below 50 Below 45 Below 40 Below 35 of the modal class and the frequency Below 30 Below 25 Below 20 Below 15 Number of families of the class succeeding the modal 97 95 90 80 class (ignoring signs). 60 30 12 4 h = class interval of the distribution. As you can see this is a case of You may note that in case of cumulative frequency distribution. In continuous series, class intervals order to calculate mode, you will have should be equal and series should be to covert it into an exclusive series. In TABLE 5.7 Grouping Table Income (in ’000 Rs) Group Frequency I I I II IV V VI 45–50 97 – 95 = 2 40–45 95 – 90 = 5 7 17 35–40 90 – 80 = 10 15 30–35 80 – 60 = 20 30 35 25–30 60 – 30 = 30 50 60 20–25 30 – 12 = 18 48 68 15–20 12 – 4 = 8 26 56 10–15 4 12 30 TABLE 5.8 Analysis Table Columns Class Intervals 45–50 40–45 35–40 30–35 25–30 20–25 15–20 10–15 I × I × × I II × × IV × × × V × × × VI × × × Total – – 1 3 6 3 1 – 7 0 STATISTICS FOR ECONOMICS this example, the series is in the • Take a small survey in your class descending order. Grouping and to know the student’s preference Analysis table would be made to for Chinese food using determine the modal class. appropriate measure of central tendency. The value of the mode lies in • Can mode be located 25–30 class interval. By inspection graphically? also, it can be seen that this is a modal class. . 6 RELATIVE POSITION OF ARITHMETIC Now L = 25, D1 = (30 – 18) = 12, D2 MEAN, MEDIAN AND MODE = (30 – 20) = 10, h = 5 Using the formula, you can obtain Suppose we express, the value of the mode as: Arithmetic Mean = Me MO (in ’000 Rs) Median = Mi Mode = Mo D1 M= h so that e, i and o are the suffixes. D1 + D2 The relative magnitude of the three are 12 M e>M i>M o or M e<M i<M o (suffixes = 25 + 5 = Rs 27,273 occurring in alphabetical order). The 10+12 median is always between the Thus the modal worker family’s arithmetic mean and the mode. monthly income is Rs 27,273. 7. CONCLUSION Activities Measures of central tendency or • A shoe company, making shoes averages are used to summarise the for adults only, wants to know data. It specifies a single most the most popular size of shoes. representative value to describe the Which average will be most appropriate for it? data set. Arithmetic mean is the most commonly used average. It is simple MEASURES OF CENTRAL TENDENCY 7 1 to calculate and is based on all the graphically. In case of open-ended observations. But it is unduly affected distribution they can also be easily by the presence of extreme items. computed. Thus, it is important to Median is a better summary for such select an appropriate average data. Mode is generally used to depending upon the purpose of describe the qualitative data. Median analysis and the nature of the and mode can be easily computed distribution. Recap • The measure of central tendency summarises the data with a single value, which can represent the entire data. • Arithmetic mean is defined as the sum of the values of all observations divided by the number of observations. • The sum of deviations of items from the arithmetic mean is always equal to zero. • Sometimes, it is important to assign weights to various items according to their importance. • Median is the central value of the distribution in the sense that the number of values less than the median is equal to the number greater than the median. • Quartiles divide the total set of values into four equal parts. • Mode is the value which occurs most frequently. EXERCISES . 1 Which average would be suitable in the following cases? i) ( Average size of readymade garments. i) (i Average intelligence of students in a class. ii (i) Average production in a factory per shift. i) (v Average wages in an industrial concern. v () When the sum of absolute deviations from average is least. v) (i When quantities of the variable are in ratios. vi (i) In case of open-ended frequency distribution. . 2 Indicate the most appropriate alternative from the multiple choices provided against each question. i) ( The most suitable average for qualitative measurement is a () arithmetic mean b ( ) median c () mode 7 2 STATISTICS FOR ECONOMICS d ( ) geometric mean e () none of the above i) (i Which average is affected most by the presence of extreme items? a () median b ( ) mode c () arithmetic mean d ( ) geometric mean e () harmonic mean ii (i) The algebraic sum of deviation of a set of n values from A.M. is a () n b () 0 c () 1 d ( ) none of the above [Ans. (i) b (ii) c (iii) b] . 3 Comment whether the following statements are true or false. i) ( The sum of deviation of items from median is zero. i) (i An average alone is not enough to compare series. ii (i) Arithmetic mean is a positional value. i) (v Upper quartile is the lowest value of top 25% of items. v () Median is unduly affected by extreme observations. [Ans. (i) False (ii) True (iii) False (iv) True (v) False] . 4 If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series: Profit per retail shop (in Rs) 0-10 10-20 20-30 30-40 40-50 50-60 Number of retail shops 12 18 27 - 17 6 (Ans. The value of missing frequency is 20 and value of the median is Rs 27.41) . 5 The following table gives the daily income of ten workers in a factory. Find the arithmetic mean. Workers A B C D E F G H I J Daily Income (in Rs) 120 150 180 200 250 300 220 350 370 260 (Ans. Rs 240) . 6 Following information pertains to the daily income of 150 families. Calculate the arithmetic mean. Income (in Rs) Number of families More than 75 150 ,, 85 140 ,, 95 115 ,, 105 95 ,, 115 70 ,, 125 60 ,, 135 40 ,, 145 25 (Ans. Rs 116.3) MEASURES OF CENTRAL TENDENCY 7 3 . 7 The size of land holdings of 380 families in a village is given below. Find the median size of land holdings. Size of Land Holdings (in acres) Less than 100 100–200 200 – 300 300–400 400 and above. – Number of families 40 89 148 64 39 (Ans. 241.22 acres) . 8 The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers. Daily Income (in Rs) 10–14 15–19 20–24 25–29 30–34 35–39 Number of workers 5 10 15 20 10 5 (Hint: compute median, lower quartile and upper quartile.) [Ans. (a) Rs 25.11 (b) Rs 19.92 (c) Rs 29.19] . 9 The following table gives production yield in kg. per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode production yield. Production yield (kg. per hectare) 50–53 53–56 56–59 59–62 62–65 65–68 68–71 71–74 74–77 Number of farms 3 8 14 30 36 28 16 10 5 (Ans. mean = 63.82 kg. per hectare, median = 63.67 kg. per hectare, mode = 63.29 kg. per hectare) CHAPTER 7 Correlation As the summer heat rises, hill Studying this chapter should stations, are crowded with more and enable you to: more visitors. Ice-cream sales become • understand the meaning of the term correlation; more brisk. Thus, the temperature is • understand the nature of related to number of visitors and sale relationship between two of ice-creams. Similarly, as the supply variables; of tomatoes increases in your local • calculate the different measures mandi, its price drops. When the local of correlation; • analyse the degree and direction harvest starts reaching the market, of the relationships. the price of tomatoes drops from a princely Rs 40 per kg to Rs 4 per kg or 1. INTRODUCTION even less. Thus supply is related to price. Correlation analysis is a means In previous chapters you have learnt for examining such relationships how to construct summary measures systematically. It deals with questions out of a mass of data and changes among similar variables. Now you will such as: learn how to examine the relationship • Is there any relationship between between two variables. two variables? 92 STATISTICS FOR ECONOMICS integral part of the theory of demand, which you will read in class XII. Low rainfall is related to low agricultural productivity. Such examples of relationship may be given a cause and effect interpretation. Others may be just coincidence. The relation between the arrival of migratory birds in a • If the value of one variable sanctuary and the birth rates in the changes, does the value of the locality can not be given any cause other also change? and ef fect interpretation. The relationships are simple coincidence. The relationship between size of the shoes and money in your pocket is another such example. Even if relationship exist, they are difficult to explain it. In another instance a third variable’s impact on two variables may give rise to a relation between the two variables. Brisk sale of ice-creams may • Do both the variables move in the be related to higher number of deaths same direction? due to drowning. The victims are not drowned due to eating of ice-creams. Rising temperature leads to brisk sale of ice-creams. Moreover, large number of people start going to swimming pools to beat the heat. This might have raised the number of deaths by drowning. Thus temperature is behind the high correlation between the sale of ice-creams and deaths due to drowning. • How strong is the relationship? What Does Correlation Measure? 2. TYPES OF RELATIONSHIP Correlation studies and measures the Let us look at various types of direction and intensity of relationship relationship. The relation between among variables. Correlation movements in quantity demanded measures covariation, not causation. and the price of a commodity is an Correlation should never be CORRELATION 93 interpreted as implying cause and 3. T E C H N I Q U E S FOR MEASURING effect relation. The presence of CORRELATION correlation between two variables X Widely used techniques for the study and Y simply means that when the of correlation are scatter diagrams, value of one variable is found to Karl Pearson’s coef ficient of change in one direction, the value of correlation and Spearman’s rank the other variable is found to change correlation. either in the same direction (i.e. A scatter diagram visually presents positive change) or in the opposite the nature of association without direction (i.e. negative change), but in giving any specific numerical value. A a definite way. For simplicity we numerical measure of linear assume here that the correlation, if relationship between two variables is it exists, is linear, i.e. the relative given by Karl Pearson’s coefficient of movement of the two variables can be correlation. A relationship is said to be linear if it can be represented by a represented by drawing a straight line straight line. Another measure is on graph paper. Spearman’s coefficient of correlation, which measures the linear association Types of Correlation between ranks assigned to indiviual Correlation is commonly classified items according to their attributes. into negative and positive correlation. Attributes are those variables which The correlation is said to be positive cannot be numerically measured such when the variables move together in as intelligence of people, physical the same direction. When the income appearance, honesty etc. rises, consumption also rises. When Scatter Diagram income falls, consumption also falls. Sale of ice-cream and temperature A scatter diagram is a useful move in the same direction. The technique for visually examining the correlation is negative when they move for m of relationship, without in opposite directions. When the price calculating any numerical value. In this technique, the values of the two of apples falls its demand increases. variables are plotted as points on a When the prices rise its demand graph paper. The cluster of points, so decreases. When you spend more time plotted, is referred to as a scatter in studying, chances of your failing diagram. From a scatter diagram, one decline. When you spend less hours can get a fairly good idea of the nature in study, chances of your failing of relationship. In a scatter diagram increase. These are instances of the degree of closeness of the scatter negative correlation. The variables points and their overall direction move in opposite direction. enable us to examine the relation- 94 STATISTICS FOR ECONOMICS ship. If all the points lie on a line, the Inspection of the scatter diagram correlation is perfect and is said to be gives an idea of the nature and unity. If the scatter points are widely intensity of the relationship. dispersed around the line, the correlation is low. The correlation is Karl Pearson’s Coef ficient of said to be linear if the scatter points Correlation lie near a line or on a line. This is also known as product moment Scatter diagrams spanning over correlation and simple correlation Fig. 7.1 to Fig. 7.5 give us an idea of coefficient. It gives a precise numerical the relationship between two value of the degree of linear variables. Fig. 7.1 shows a scatter relationship between two variables X around an upward rising line and Y. The linear relationship may be given by indicating the movement of the Y = a + bX variables in the same direction. When This type of relation may be X rises Y will also rise. This is positive described by a straight line. The correlation. In Fig. 7.2 the points are intercept that the line makes on the found to be scattered around a Y-axis is given by a and the slope of downward sloping line. This time the the line is given by b. It gives the variables move in opposite directions. change in the value of Y for very small When X rises Y falls and vice versa. change in the value of X. On the other This is negative correlation. In Fig.7.3 hand, if the relation cannot be there is no upward rising or downward represented by a straight line as in sloping line around which the points Y = X2 are scattered. This is an example of the value of the coefficient will be zero. no correlation. In Fig. 7.4 and Fig. 7.5 It clearly shows that zero correlation the points are no longer scattered need not mean absence of any type around an upward rising or downward of relation between the two variables. falling line. The points themselves are Let X1, X2, ..., XN be N values of X on the lines. This is referred to as and Y1, Y2 ,..., YN be the corresponding perfect positive correlation and perfect values of Y. In the subsequent negative correlation respectively. presentations the subscripts indicating the unit are dropped for the Activity sake of simplicity. The arithmetic means of X and Y are defined as • Collect data on height, weight ΣX ΣY and marks scored by students in your class in any two subjects X= ; Y= N N in class X. Draw the scatter and their variances are as follows diagram of these variables taking two at a time. What type of Σ( X - X )2 ΣX 2 relationship do you find? s2 x = = - X2 N N CORRELATION 95 96 STATISTICS FOR ECONOMICS Properties of Correlation Coefficient Σ( Y - Y )2 ΣY 2 and s 2 y = = - Y2 Let us now discuss the properties of N N the correlation coefficient The standard deviations of X and • r has no unit. It is a pure number. Y respectively are the positive square It means units of measurement are roots of their variances. Covariance of not part of r. r between height in X and Y is defined as feet and weight in kilograms, for instance, is 0.7. Σ( X - X )( Y - Y ) Σxy Cov(X,Y) = = • A negative value of r indicates an N N inverse relation. A change in one Where x = X - X and y = X - Y variable is associated with change in the other variable in the are the deviations of the ith value of X opposite direction. When price of and Y from their mean values a commodity rises, its demand respectively. falls. When the rate of interest The sign of covariance between X rises the demand for funds also and Y determines the sign of the falls. It is because now funds have correlation coefficient. The standard become costlier. deviations are always positive. If the covariance is zero, the correlation coefficient is always zero. The product moment correlation or the Karl Pearson’s measure of correlation is given by r = Σxy Ns s ...(1) x y or Σ( X - X ) ( Y - Y ) r= ...(2) Σ( X - X )2 Σ( Y - Y )2 or (ΣX )(ΣY ) ΣXY - • If r is positive the two variables r= N move in the same direction. When (ΣX ) 2 (ΣY ) 2 ...(3) the price of coffee, a substitute of ΣX 2 - ΣY 2 - N N tea, rises the demand for tea also rises. Improvement in irrigation or NΣXY (ΣX )(ΣY ) facilities is associated with higher r= yield. When temperature rises the NΣX 2 (ΣX )2 NΣY 2 (ΣY )2 ...(4) sale of ice-creams becomes brisk. CORRELATION 97 • If r = 0 the two variables are before correlation is calculated. An uncorrelated. There is no linear epidemic spreads in some villages and relation between them. However the gover nment sends a team of other types of relation may be doctors to the affected villages. The there. correlation between the number of • If r = 1 or r = –1 the correlation is deaths and the number of doctors sent perfect. The relation between them to the villages is found to be positive. is exact. Normally the health care facilities • A high value of r indicates strong provided by the doctors are expected linear relationship. Its value is to reduce the number of deaths said to be high when it is close to showing a negative correlation. This +1 or –1. happened due to other reasons. The • A low value of r indicates a weak data relate to a specific time period. linear relation. Its value is said to Many of the reported deaths could be be low when it is close to zero. terminal cases where the doctors • The value of the correlation coefficient lies between minus one could do little. Moreover, the benefit and plus one, –1 ≤ r ≤ 1. If, in of the presence of doctors becomes visible after some time. It is also any exercise, the value of r is outside this range it indicates error possible that the reported deaths are in calculation. not due to the epidemic. A tsunami • The value of r is unaffected by the suddenly hits the state and death toll change of origin and change of rises. scale. Given two variables X and Y Let us illustrate the calculation of let us define two new variables. r by examining the relationship between years of schooling of the X A Y C farmer and the annual yield per acre. U= ; V= B D where A and C are assumed means of Example 1 X and Y respectively. B and D are common factors. Then No. of years Annual yield per of schooling acre in ’000 (Rs) rxy = ruv of farmers This. property is used to calculate 0 4 correlation coefficient in a highly 2 4 4 6 simplified manner, as in the step 6 10 deviation method. 8 10 As you have read in chapter 1, the 10 8 statistical methods are no substitute 12 7 for common sense. Here, is another Formula 1 needs the value of example, which highlights the need for understanding the data properly Σxy, s x , s y 98 STATISTICS FOR ECONOMICS From Table 7.1 we get, education, higher will be the yield per acre. It underlines the importance of Σxy = 42, farmers’ education. To use formula (3) Σ( X - X )2 112 sx = = , N 7 (ΣX )(ΣY ) ΣXY - r= N Σ( Y - Y )2 38 (ΣX ) 2 (ΣY ) 2 ...(3) sy = = ΣX 2 - ΣY 2 - N 7 N N Substituting these values in the value of the following expressions formula (1) have to be calculated i.e. 42 ΣXY, ΣX 2 , ΣY 2 . r= = 0.644 112 38 Now apply formula (3) to get the 7 value of r. 7 7 The same value can be obtained Let us know the interpretation of from formula (2) also. different values of r. The correlation coefficient between marks secured in Σ ( X - X )( Y - Y ) English and Statistics is, say, 0.1. It r= ...(2) Σ ( X - X )2 Σ ( Y - Y )2 means that though the marks secured in the two subjects are positively 42 correlated, the strength of the r= = 0.644 112 38 relationship is weak. Students with high Thus years of education of the marks in English may be getting farmers and annual yield per acre are relatively low marks in statistics. Had positively correlated. The value of r is the value of r been, say, 0.9, students also large. It implies that more the with high marks in English will number of years farmers invest in invariably get high marks in Statistics. TABLE 7.1 Calculation of r between years of schooling of farmers and annual yield Years of (X– X ) (X– X ) 2 Annual yield (Y– Y ) (Y– Y )2 (X– X )(Y– Y ) Education per acre in ’000 Rs (X) (Y) 0 –6 36 4 –3 9 18 2 –4 16 4 –3 9 12 4 –2 4 6 –1 1 2 6 0 0 10 3 9 0 8 2 4 10 3 9 6 10 4 16 8 1 1 4 12 6 36 7 0 0 0 Σ X=42 Σ (X– X )2=112 Σ Y=49 Σ (Y– Y )2=38 Σ (X– X )(Y– Y )=42 CORRELATION 99 An example of negative correlation TABLE 7.2 is the relation between arrival of Year Annual growth Gross Domestic vegetables in the local mandi and price of National Saving as Income percentage of GDP of vegetables. If r is –0.9, vegetable 1992–93 14 24 supply in the local mandi will be 1993–94 17 23 accompanied by lower price of 1994–95 18 26 vegetables. Had it been –0.1 large 1995–96 17 27 1996–97 16 25 vegetable supply will be accompanied 1997–98 12 25 by lower price, not as low as the price, 1998–99 16 23 when r is –0.9. The extent of price fall 1999–00 11 25 2000–01 8 24 depends on the absolute value of r. 2001–02 10 23 Had it been zero there would have been no fall in price, even after large Source: Economic Survey, (2004–05) Pg. 8,9 supplies in the market. This is also a a pr operty of r. It is that r is possibility if the increase in supply is independent of change in origin and taken care of by a good transport scale. It is also known as step network transferring it to other deviation method. It involves the markets. transformation of the variables X and Y as follows: Activity X A Y B U= ;V = • Look at the following table. h k Calculate r between annual where A and B are assumed means, h growth of national income at and k are common factors. current price and the Gross Then rUV = rXY Domestic Saving as percentage This can be illustrated with the of GDP. exercise of analysing the correlation between price index and money Step deviation method to calculate supply. correlation coefficient. Example 2 When the values of the variables are large, the burden of calculation Price 120 150 190 220 230 index (X) can be considerably reduced by using Money 1800 2000 2500 2700 3000 a pr operty of r. It is that r is supply independent of change in origin and in Rs crores (Y) scale. It is also known as step The simplification, using step deviation method. It involves the deviation method is illustrated below. transformation of the variables X and Let A = 100; h = 10; B = 1700 and Y as follows: k = 100 100 STATISTICS FOR ECONOMICS The table of transformed variables Activity is as follows: • Take some examples of India’s Calculation of r between price population and national income. index and money supply using step Calculate the corr elation between them using step deviation method deviation method and see the simplification. TABLE 7.3 U V Spearman’s rank correlation Ê X - 100 ˆ Ê Y - 1700 ˆ Spearman’s rank correlation was Á Ë 10 ˜ Á 100 ˜ ¯ Ë ¯ U2 V2 UV developed by the British psychologist 2 1 4 1 2 C.E. Spearman. It is used when the 5 3 25 9 15 variables cannot be measured 9 8 81 64 72 meaningfully as in the case of price, 12 10 144 100 120 income, weight etc. Ranking may be 13 13 169 169 169 more meaningful when the measurements of the variables are ΣU = 41; ΣV = 35; ΣU 2 = 423; suspect. Consider the situation where ΣV 2 = 343; ΣUV = 378 we are required to calculate the Substituting these values in formula correlation between height and weight (3) of students in a remote village. Neither measuring rods nor weighing scales (ΣU )(ΣV ) are available. The students can be ΣUV - r= N easily ranked in terms of height and (ΣU ) 2 (ΣV )2 (3) weight without using measuring rods ΣU 2 - ΣV 2 - and weighing scales. N N There are also situations when you are required to quantify qualities such 41 ¥ 35 378 - as fairness, honesty etc. Ranking may = 5 be a better alternative to quantifica- (41) 2 (35) 2 tion of qualities. Moreover, sometimes 423 - 343 - 5 5 the correlation coefficient between two variables with extreme values may be = 0.98 quite different from the coefficient without the extreme values. Under This strong positive correlation these circumstances rank correlation between price index and money provides a better alternative to simple supply is an important premise of correlation. monetary policy. When the money Rank correlation coefficient and supply grows the price index also simple correlation coefficient have the rises. same interpretation. Its formula has CORRELATION 101 been derived from simple correlation concerning the data is not utilised. coefficient where individual values The first differences of the values of have been replaced by ranks. These the items in the series, arranged in ranks are used for the calculation of order of magnitude, are almost never correlation. This coefficient provides constant. Usually the data cluster a measure of linear association around the central values with smaller between ranks assigned to these differences in the middle of the array. units, not their values. It is the If the first differences were constant Product Moment Correlation between then r and r k would give identical the ranks. Its formula is results. The first difference is the difference of consecutive values. 6ΣD 2 rk = 1 ...(4) Rank correlation is preferred to n3 n Pearsonian coefficient when extreme where n is the number of observations values are present. In general and D the deviation of ranks assigned rk is less than or equal to r. to a variable from those assigned to The calculation of rank correlation the other variable. When the ranks are will be illustrated under three repeated the formula is situations. rk = 1– 1. The ranks are given. 2. The ranks are not given. They have È ( m 31 - m1 ) ( m 32 - m 2 ) ˘ 6 ÍΣD2 + + + ...˙ to be worked out from the data. Î 12 12 ˚ 3. Ranks are repeated. n( n 2 - 1) where m1, m2, ..., are the number of Case 1: When the ranks are given m 31 m1 Example 3 repetitions of ranks and ..., 12 Five persons are assessed by three their corresponding correction judges in a beauty contest. We have factors. This correction is needed for to find out which pair of judges has every repeated value of both variables. the nearest approach to common If three values are repeated, there will perception of beauty. be a correction for each value. Every Competitors time m1 indicates the number of times Judge 1 2 3 4 5 a value is repeated. All the properties of the simple A 1 2 3 4 5 B 2 4 1 5 3 correlation coefficient are applicable C 1 3 5 2 4 here. Like the Pearsonian Coefficient of correlation it lies between 1 and There are 3 pairs of judges –1. However, generally it is not as necessitating calculation of rank accurate as the ordinary method. This correlation thrice. Formula (4) will be is due the fact that all the information used — 102 STATISTICS FOR ECONOMICS 6ΣD2 Case 2: When the ranks are not given rs = 1 - ...(4) n3 - n Example 4 The rank correlation between A and B is calculated as follows: We are given the percentage of marks, secured by 5 students in Economics and Statistics. Then the ranking has A B D D2 to be worked out and the rank 1 2 –1 1 correlation is to be calculated. 2 4 –2 4 3 1 2 4 4 5 –1 1 Student Marks in Marks in 5 3 2 4 Statistics Economics (X) (Y) Total 14 A 85 60 B 60 48 Substituting these values in C 55 49 formula (4) D 65 50 E 75 55 6ΣD2 rs = 1 - ...(4) n3 - n Student Ranking in Ranking in 6 ¥ 14 84 Statistics Economics =1- =1- = 1 - 0.7 = 0.3 (Rx) (RY ) 5 -5 3 120 A 1 1 The rank correlation between A B 4 5 and C is calculated as follows: C 5 4 D 3 3 E 2 2 A C D D2 1 1 0 0 Once the ranking is complete 2 3 –1 1 formula (4) is used to calculate rank 3 5 –2 4 correlation. 4 2 2 4 5 4 1 1 Case 3: When the ranks are repeated Total 10 Example 5 Substituting these values in The values of X and Y are given as formula (4) the rank correlation is 0.5. X 25 45 35 40 15 19 35 42 Similarly, the rank correlation Y 55 60 30 35 40 42 36 48 between the rankings of judges B and In order to work out the rank C is 0.9. Thus, the perceptions of correlation, the ranks of the values judges A and C are the closest. Judges are worked out. Common ranks are B and C have very different tastes. given to the repeated items. The CORRELATION 103 common rank is the mean of the ranks m 3 - m 23 - 2 1 which those items would have = = 12 12 2 assumed if they were slightly different Using this equation from each other. The next item will be assigned the rank next to the rank È (m3 - m ) ˘ 6 ÍΣD 2 + ˙ rs = 1 - Î ˚ ...(5) already assumed. The formula of 12 Spear man’s rank correlation n 3 - n coef ficient when the ranks are Substituting the values of these repeated is as follows expressions rs = 1 - 6(65.5 + 0.5) 396 rs = 1 - =1- È ( m - m1 ) ( m 2 - m 2 ) 3 3 ˘ 83 - 8 504 6 ÍΣD2 + + + ...˙ 1 Î 12 12 ˚ = 1 - 0.786 = 0.214 n( n 2 - 1) Thus there is positive rank correlation where m1, m2, ..., are the number between X and Y. Both X and Y move of r epetitions of ranks and in the same direction. However, the relationship cannot be described as m 31 - m1 strong. ..., their corresponding 12 correction factors. Activity X has the value 35 both at the • Collect data on marks scored by 4th and 5th rank. Hence both are 10 of your classmates in class given the average rank i.e., IX and X examinations. Calculate the rank correlation coefficient 4+5 between them. If your data do not th = 4.5 th rank 2 have any repetition, repeat the exercise by taking a data set having repeated ranks. What are X Y Rank of Rank of Deviation in D2 the circumstances in which rank Ranking corr elation coef ficient is XR' YR'' D=R'–R'' preferred to simple correlation 25 55 6 2 4 16 coefficient? If data are precisely 45 80 1 1 0 0 measured will you still prefer 35 30 4.5 8 3.5 12.25 rank correlation coefficient to 40 35 3 7 –4 16 simple correlation? When can 15 40 8 5 3 9 you be indifferent to the choice? 19 42 7 4 3 9 Discuss in class. 35 36 4.5 6 –1.5 2.25 42 48 2 3 –1 1 4. CONCLUSION Total ΣD = 65.5 We have discussed some techniques The necessary correction thus is for studying the relationship between 104 STATISTICS FOR ECONOMICS two variables, particularly the linear relationship. When the variables relationship. The scatter diagram gives cannot be measured precisely, rank a visual presentation of the correlation can meaningfully be used. relationship and is not confined to These measures however do not imply linear relations. Measures of causation. The knowledge of correlation such as Karl Pearson’s correlation gives us an idea of the coefficient of corr elation and direction and intensity of change in a Spearman’s rank correlation are variable when the correlated variable strictly the measures of linear changes. Recap • Correlation analysis studies the relation between two variables. • Scatter diagrams give a visual presentation of the nature of relationship between two variables. • Karl Pearson’s coefficient of correlation r measures numerically only linear relationship between two variables. r lies between –1 and 1. • When the variables cannot be measured precisely Spearman’s rank correlation can be used to measure the linear relationship numerically. • Repeated ranks need correction factors. • Correlation does not mean causation. It only means covariation. EXERCISES 1. The unit of correlation coefficient between height in feet and weight in kgs is (i) kg/feet (ii) percentage (iii) non-existent 2. The range of simple correlation coefficient is (i) 0 to infinity (ii) minus one to plus one (iii) minus infinity to infinity 3. If rxy is positive the relation between X and Y is of the type (i) When Y increases X increases (ii) When Y decreases X increases (iii) When Y increases X does not change CORRELATION 105 4. If rxy = 0 the variable X and Y are (i) linearly related (ii) not linearly related (iii) independent 5. Of the following three measures which can measure any type of relationship (i) Karl Pearson’s coefficient of correlation (ii) Spearman’s rank correlation (iii) Scatter diagram 6. If precisely measured data are available the simple correlation coefficient is (i) more accurate than rank correlation coefficient (ii) less accurate than rank correlation coefficient (iii) as accurate as the rank correlation coefficient 7. Why is r preferred to covariance as a measure of association? 8. Can r lie outside the –1 and 1 range depending on the type of data? 9. Does correlation imply causation? 10. When is rank correlation more precise than simple correlation coefficient? 11. Does zero correlation mean independence? 12. Can simple correlation coefficient measure any type of relationship? 13. Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result. 14. Measure the height of your classmates. Ask them the height of their benchmate. Calculate the correlation coefficient of these two variables. Interpret the result. 15. List some variables where accurate measurement is difficult. 16. Interpret the values of r as 1, –1 and 0. 17. Why does rank correlation coefficient differ from Pearsonian correlation coefficient? 18. Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y) X 65 66 57 67 68 69 70 72 Y 67 56 65 68 72 72 69 71 (Ans. r = 0.603) 19. Calculate the correlation coefficient between X and Y and comment on their relationship: X –3 –2 –1 1 2 3 Y 9 4 1 1 4 9 (Ans. r = 0) 106 STATISTICS FOR ECONOMICS 20. Calculate the correlation coefficient between X and Y and comment on their relationship X 1 3 4 5 7 8 Y 2 6 8 10 14 16 (Ans. r = 1) Activity • Use all the formulae discussed here to calculate r between India’s national income and export taking at least ten observations. CHAPTER Measures of Dispersion measures, which seek to quantify Studying this chapter should variability of the data. enable you to: • know the limitations of averages; Three friends, Ram, Rahim and • appreciate the need of measures Maria are chatting over a cup of tea. of dispersion; During the course of their • enumerate various measures of conversation, they start talking about dispersion; their family incomes. Ram tells them • calculate the measures and that there are four members in his compare them; family and the average income per • distinguish between absolute member is Rs 15,000. Rahim says that and relative measures. the average income is the same in his family, though the number of members 1. INTRODUCTION is six. Maria says that there are five members in her family, out of which In the previous chapter, you have one is not working. She calculates that studied how to sum up the data into the average income in her family too, a single representative value. However, is Rs 15,000. They are a little surprised that value does not reveal the since they know that Maria’s father is variability present in the data. In this earning a huge salary. They go into chapter you will study those details and gather the following data: MEASURES OF DISPERSION 75 Family Incomes variation in values, your understan- Sl. No. Ram Rahim Maria ding of a distribution improves 1. 12,000 7,000 0 considerably. For example, per capita 2. 14,000 10,000 7,000 income gives only the average income. 3. 16,000 14,000 8,000 A measure of dispersion can tell you 4. 18,000 17,000 10,000 about income inequalities, thereby 5. ----- 20,000 50,000 6. ----- 22,000 ------ improving the understanding of the relative standards of living enjoyed by Total income 60,000 90,000 75,000 Average income 15,000 15,000 15,000 different strata of society. Dispersion is the extent to which Do you notice that although the values in a distribution differ from the average is the same, there are average of the distribution. considerable differences in individual To quantify the extent of the incomes? variation, there are certain measures It is quite obvious that averages namely: try to tell only one aspect of a (i) Range distribution i.e. a representative size (ii) Quartile Deviation of the values. To understand it better, (iii) Mean Deviation you need to know the spread of values (iv) Standard Deviation also. Apart from these measures which You can see that in Ram’s family., give a numerical value, there is a dif ferences in incomes are graphic method for estimating comparatively lower. In Rahim’s dispersion. family, differences are higher and in Range and Quartile Deviation Maria’s family are the highest. measure the dispersion by calculating Knowledge of only average is the spread within which the values lie. insufficient. If you have another value Mean Deviation and Standard which reflects the quantum of Deviation calculate the extent to which the values differ from the average. 2. MEASURES BASED UPON SPREAD OF VALUES Range Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus, R=L–S Higher value of Range implies higher dispersion and vice-versa. 76 STATISTICS FOR ECONOMICS Activities Quartile Deviation Look at the following values: The presence of even one extremely 20, 30, 40, 50, 200 high or low value in a distribution can • Calculate the Range. reduce the utility of range as a • What is the Range if the value measure of dispersion. Thus, you may 200 is not present in the data need a measure which is not unduly set? • If 50 is replaced by 150, what affected by the outliers. will be the Range? In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we Range: Comments Range is unduly affected by extreme get the values of Quartiles and values. It is not based on all the Median. (You have already read about values. As long as the minimum and these in Chapter 5). maximum values remain unaltered, The upper and lower quartiles (Q3 any change in other values does not and Q 1, respectively) are used to affect range. It can not be calculated calculate Inter Quartile Range which for open-ended frequency distri- is Q3 – Q1. bution. Inter -Quartile Range is based Notwithstanding some limitations, upon middle 50% of the values in a Range is understood and used distribution and is, therefore, not frequently because of its simplicity. affected by extreme values. Half of For example, we see the maximum the Inter -Quartile Range is called and minimum temperatures of Quartile Deviation. Thus: different cities almost daily on our TV Q3 - Q1 screens and form judgments about the Q .D . = 2 temperature variations in them. Q.D. is therefore also called Semi- Open-ended distributions are those Inter Quartile Range. in which either the lower limit of the lowest class or the upper limit of the Calculation of Range and Q.D. for highest class or both are not ungrouped data specified. Example 1 Activity Calculate Range and Q.D. of the following observations: • Collect data about 52-week 20, 25, 29, 30, 35, 39, 41, high/low of 10 shares from a newspaper. Calculate the range 48, 51, 60 and 70 of share prices. Which stock is Range is clearly 70 – 20 = 50 most volatile and which is the For Q.D., we need to calculate most stable? values of Q3 and Q1. MEASURES OF DISPERSION 77 n +1 Range is just the dif ference Q1 is the size of th value. between the upper limit of the highest 4 class and the lower limit of the lowest n being 11, Q1 is the size of 3rd class. So Range is 90 – 0 = 90. For value. Q.D., first calculate cumulative As the values are already arranged frequencies as follows: in ascending order, it can be seen that Q1, the 3rd value is 29. [What will you Class- Frequencies Cumulative Intervals Frequencies do if these values are not in an order?] CI f c. f. 3( n + 1) 0–10 5 05 Similarly, Q3 is size of th 10–20 8 13 4 20–40 16 29 value; i.e. 9th value which is 51. Hence 40–60 7 36 Q3 = 51 60–90 4 40 Q3 - Q1 51 - 29 n = 40 Q .D . = = = 11 2 2 n th Do you notice that Q.D. is the Q1 is the size of value in a 4 average difference of the Quartiles continuous series. Thus it is the size from the median. of the 10th value. The class containing Activity the 10th value is 10–20. Hence Q1 lies • Calculate the median and check in class 10–20. Now, to calculate the whether the above statement is exact value of Q 1 , the following correct. formula is used: Calculation of Range and Q.D. for a n cf frequency distribution. Q1 = L + 4 ·i f Example 2 Where L = 10 (lower limit of the For the following distribution of marks relevant Quartile class) scored by a class of 40 students, c.f. = 5 (Value of c.f. for the class calculate the Range and Q.D. preceding the Quartile class) TABLE 6.1 i = 10 (interval of the Quartile Class intervals No. of students class), and CI (f) f = 8 (frequency of the Quartile 0–10 5 class) Thus, 10–20 8 10 - 5 20–40 16 Q1 = 10 + · 10 = 16.25 40–60 7 8 60–90 4 3n th 40 Similarly, Q3 is the size of 4 78 STATISTICS FOR ECONOMICS value; i.e., 30th value, which lies in to rich and poor, from the median of class 40–60. Now using the formula the entire group. for Q3, its value can be calculated as Quartile Deviation can generally be follows: calculated for open-ended distribu- tions and is not unduly affected by 3n - c.f. extreme values. Q3 = L + 4 i f 3. M EASURES OF D ISPERSION FROM 30 - 29 AVERAGE Q3 = 40 + 20 7 Recall that dispersion was defined as Q3 = 42.87 the extent to which values differ from their average. Range and Quartile 42.87 - 16.25 Deviation do not attempt to calculate, Q.D. = = 13.31 2 how far the values are, from their In individual and discrete series, Q1 average. Yet, by calculating the spread of values, they do give a good idea n +1 th about the dispersion. Two measures is the size of value, but in a 4 which are based upon deviation of the continuous distribution, it is the size values from their average are Mean n th Deviation and Standard Deviation. of value. Similarly, for Q3 and Since the average is a central 4 value, some deviations are positive median also, n is used in place of n+1. and some are negative. If these are added as they are, the sum will not reveal anything. In fact, the sum of If the entire group is divided into deviations from Arithmetic Mean is two equal halves and the median calculated for each half, you will have always zero. Look at the following two the median of better students and the sets of values. median of weak students. These Set A : 5, 9, 16 medians differ from the median of the Set B : 1, 9, 20 entire group by 13.31 on an average. You can see that values in Set B Similarly, suppose you have data about incomes of people of a town. are farther from the average and hence Median income of all people can be more dispersed than values in Set A. calculated. Now if all people are Calculate the deviations from divided into two equal groups of rich Arithmetic Mean amd sum them up. and poor, medians of both groups can What do you notice? Repeat the same be calculated. Quartile Deviation will with Median. Can you comment upon tell you the average difference between the quantum of variation from the medians of these two groups belonging calculated values? MEASURES OF DISPERSION 79 Mean Deviation tries to overcome Mean Deviation which is simply the this problem by ignoring the signs of arithmetic mean of the differences of deviations, i.e., it considers all the values from their average. The deviations positive. For standard average used is either the arithmetic deviation, the deviations are first mean or median. squared and averaged and then (Since the mode is not a stable square root of the average is found. average, it is not used to calculate We shall now discuss them separately Mean Deviation.) in detail. Activities • Calculate the total distance to be Mean Deviation travelled by students if the Suppose a college is proposed for college is situated at town A, at students of five towns A, B, C, D and town C, or town E and also if it E which lie in that order along a road. is exactly half way between A and Distances of towns in kilometres from E. • Decide where, in you opinion, town A and number of students in the college should be establi- these towns are given below: shed, if there is only one student in each town. Does it change Town Distance No. from town A of Students your answer? A 0 90 Calculation of Mean Deviation from B 2 150 C 6 100 Arithmetic Mean for ungrouped D 14 200 data. E 18 80 Direct Method 620 Steps: Now, if the college is situated in (i) The A.M. of the values is calculated town A, 150 students from town B will (ii) Difference between each value and have to travel 2 kilometers each (a the A.M. is calculated. All total of 300 kilometres) to reach the dif ferences are considered college. The objective is to find a positive. These are denoted as |d| location so that the average distance (iii) The A.M. of these dif ferences travelled by students is minimum. (called deviations) is the Mean You may observe that the students Deviation. will have to travel more, on an average, S |d| if the college is situated at town A or i.e. M.D. = n E. If on the other hand, it is somewhere in the middle, they are Example 3 likely to travel less. The average Calculate the Mean Deviation of the distance travelled is calculated by following values; 2, 4, 7, 8 and 9. 80 STATISTICS FOR ECONOMICS SX Where Σ |d| is the sum of absolute The A.M. = =6 deviations taken from the assumed n mean. X |d| x is the actual mean. 2 4 A x is the assumed mean used to 4 2 calculate deviations. 7 1 Σ fB is the number of values below the 8 2 actual mean including the actual 9 3 mean. 12 Σ fA is the number of values above the 12 actual mean. M.D.( X ) = = 2.4 Substituting the values in the 5 above formula: Assumed Mean Method 11 + (6 - 7)(2 - 3) 12 M.D.( x ) = = = 2.4 Mean Deviation can also be calculated 5 5 by calculating deviations from an assumed mean. This method is Mean Deviation from median for adopted especially when the actual ungrouped data. mean is a fractional number. (Take care that the assumed mean is close Direct Method to the true mean). Using the values in example 3, M.D. For the values in example 3, from the Median can be calculated as suppose value 7 is taken as assumed follows, mean, M.D. can be calculated as (i) Calculate the median which is 7. under: (ii) Calculate the absolute deviations from median, denote them as |d|. Example 4 (iii) Find the average of these absolute X |d| deviations. It is the Mean Deviation. 2 5 4 3 Example 5 7 0 [X-Median] 8 1 X |d| 9 2 2 5 11 4 3 In such cases, the following 7 0 formula is used, 8 1 S| d | + ( x - Ax )(S f B - S f A ) 9 2 M.D.( x ) = 11 n MEASURES OF DISPERSION 81 M. D. from Median is thus, (iii) Multiply each |d| value with its corresponding frequency to get S | d | 11 f|d| values. Sum them up to get M.D.( median ) = = = 2.2 n 5 Σ f|d|. (iv) Apply the following formula, Short-cut method S f |d| To calculate Mean Deviation by short M.D. ( x ) = Sf cut method a value (A) is used to calculate the deviations and the Mean Deviation of the distribution following formula is applied. in Table 6.2 can be calculated as follows: M.D.( Median ) S | d| + ( Median - A )(S f B - S f A ) Example 6 = n C.I. f m.p. |d| f|d| where, A = the constant from which 10–20 5 15 25.5 127.5 deviations are calculated. (Other 20–30 8 25 15.5 124.0 notations are the same as given in the 30–50 16 40 0.5 8.0 assumed mean method). 50–70 8 60 19.5 156.0 70–80 3 75 34.5 103.5 Mean Deviation from Mean for 40 519.0 Continuous distribution S f | d | 519 M.D.( x ) = = = 12.975 TABLE 6.2 Sf 40 Profits of Number of companies Companies Mean Deviation from Median (Rs in lakhs) frequencies Class-intervals TABLE 6.3 10–20 5 Class intervals Frequencies 20–30 8 20–30 5 30–50 16 30–40 10 50–70 8 40–60 20 70–80 3 60–80 9 40 80–90 6 50 Steps: The procedure to calculate Mean (i) Calculate the mean of the Deviation from the median is the distribution. same as it is in case of M.D. from (ii) Calculate the absolute deviations Mean, except that deviations are to |d| of the class midpoints from the be taken from the median as given mean. below: 82 STATISTICS FOR ECONOMICS Example 7 Calculation of Standard Deviation for ungrouped data C.I. f m.p. |d| f|d| Four alternative methods are available 20–30 5 25 25 125 30–40 10 35 15 150 for the calculation of standard 40–60 20 50 0 0 deviation of individual values. All 60–80 9 70 20 180 these methods result in the same 80–90 6 85 35 210 value of standard deviation. These are: 50 665 (i) Actual Mean Method S f |d| (ii) Assumed Mean Method M.D.( Median ) = Sf (iii) Direct Method (iv) Step-Deviation Method 665 = = 13.3 Actual Mean Method: 50 Suppose you have to calculate the Mean Deviation: Comments standard deviation of the following Mean Deviation is based on all values: values. A change in even one value 5, 10, 25, 30, 50 will affect it. It is the least when calculated from the median i.e., it Example 8 will be higher if calculated from the mean. However it ignores the signs X d d2 of deviations and cannot be 5 –19 361 calculated for open-ended distribu- 10 –14 196 tions. 25 +1 1 30 +6 36 50 +26 676 Standard Deviation 0 1270 Standard Deviation is the positive Following formula is used: square root of the mean of squared deviations from mean. So if there are S d2 s= five values x1, x2, x3, x4 and x5, first n their mean is calculated. Then deviations of the values from mean are 1270 s= = 254 = 15.937 calculated. These deviations are then 5 squared. The mean of these squared Do you notice the value from which deviations is the variance. Positive deviations have been calculated in the square root of the variance is the above example? Is it the Actual Mean? standard deviation. (Note that Standard Deviation is Assumed Mean Method calculated on the basis of the mean For the same values, deviations may only). be calculated from any arbitrary value MEASURES OF DISPERSION 83 A x such that d = X – A x . Taking A x (This amounts to taking deviations = 25, the computation of the standard from zero) deviation is shown below: Following formula is used. Example 9 S x2 s= - ( x )2 n X d d2 4150 5 –20 400 or s = - (24 )2 10 –15 225 5 25 0 0 30 +5 25 or s = 254 = 15.937 50 +25 625 Standard Deviation is not affected –5 1275 by the value of the constant from which deviations are calculated. The Formula for Standard Deviation value of the constant does not figure 2 in the standard deviation formula. S d2 Sd Thus, Standard Deviation is s= - n Łn ł Independent of Origin. 2 1275 -5 Step-deviation Method s= - = 254 = 15.937 5 Ł5 ł If the values are divisible by a common factor, they can be so divided and The sum of deviations from a value other than actul mean is not equal standard deviation can be calculated to zero from the resultant values as follows: Example 11 Direct Method Since all the five values are divisible Standard Deviation can also be by a common factor 5, we divide and calculated from the values directly, get the following values: i.e., without taking deviations, as shown below: x x' d d2 5 1 –3.8 14.44 Example 10 10 2 –2.8 7.84 25 5 +0.2 0.04 X x2 30 6 +1.2 1.44 50 10 +5.2 27.04 5 25 10 100 0 50.80 25 625 (Steps in the calculation are same 30 900 50 2500 as in actual mean method). The following formula is used to 120 4150 calculate standard deviation: 84 STATISTICS FOR ECONOMICS S d2 Standard Deviation is not s= ·c independent of scale. Thus, if the n values or deviations are divided by x a common factor, the value of the x’ = common factor is used in the c formula to get the value of Standard c = common factor Deviation. Substituting the values, 50.80 Standard Deviation in Continuous s= 5 frequency distribution: 5 Like ungrouped data, S.D. can be s = 10.16 · 5 calculated for grouped data by any of s = 15.937 the following methods: (i) Actual Mean Method Alternatively, instead of dividing (ii) Assumed Mean Method the values by a common factor, the (iii) Step-Deviation Method deviations can be divided by a common factor. Standard Deviation Actual Mean Method can be calculated as shown below: For the values in Table 6.2, Standard Example 12 Deviation can be calculated as follows: x d d' d2 Example 13 5 –20 –4 16 10 –15 –3 9 (1) (2) (3) (4) (5) (6) (7) 25 0 0 0 CI f m fm d fd fd2 30 +5 +1 1 50 +25 +5 25 10–20 5 15 75 –25.5 –127.5 3251.25 20–30 8 25 200 –15.5 –124.0 1922.00 –1 51 30–50 16 40 640 –0.5 –8.0 4.00 50–70 8 60 480 +19.5 +156.0 3042.00 Deviations have been calculated 70–80 3 75 225 +34.5 +103.5 3570.75 from an arbitrary value 25. Common 40 1620 0 11790.00 factor of 5 has been used to divide deviations. Following steps are required: 1. Calculate the mean of the 2 S d ’2 Sd’ distribution. s= ·c n Ł n ł Sfm 1620 x= = = 40.5 Sf 40 2 51 -1 2. Calculate deviations of mid-values s= - ·5 from the mean so that 5 Ł5 ł d = m - x (Col. 5) s = 10.16 · 5 = 15.937 3. Multiply the deviations with their MEASURES OF DISPERSION 85 corresponding frequencies to get 4. Multiply ‘fd’ values (Col. 5) with ‘d’ ‘fd’ values (col. 6) [Note that Σ fd values (col. 4) to get fd2 values (col. = 0] 6). Find Σ fd2. 4. Calculate ‘fd 2 ’ values by 5. Standard Deviation can be multiplying ‘fd’ values with ‘d’ calculated by the following values. (Col. 7). Sum up these to formula. get Σ fd2. 2 Sfd2 Sfd 5. Apply the formula as under: s= - n Ł n ł Sfd2 11790 s= = = 17.168 2 n 40 11800 20 or s = - 40 Ł40 ł Assumed Mean Method or s = 294.75 = 17.168 For the values in example 13, standard deviation can be calculated Step-deviation Method by taking deviations from an assumed In case the values of deviations are mean (say 40) as follows: divisible by a common factor, the calculations can be simplified by the Example 14 step-deviation method as in the (1) (2) (3) (4) (5) (6) following example. CI f m d fd fd2 10–20 5 15 -25 –125 3125 Example 15 20–30 8 25 -15 –120 1800 30–50 16 40 0 0 0 (1) (2) (3) (4) (5) (6) (7) 50–70 8 60 +20 160 3200 CI f m d d' fd' fd'2 70–80 3 75 +35 105 3675 10–20 5 15 –25 –5 –25 125 40 +20 11800 20–30 8 25 –15 –3 –24 72 30–50 16 40 0 0 0 0 The following steps are required: 50–70 8 60 +20 +4 +32 128 1. Calculate mid-points of classes 70–80 3 75 +35 +7 +21 147 (Col. 3) 40 +4 472 2. Calculate deviations of mid-points from an assumed mean such that Steps required: d = m – A x (Col. 4). Assumed 1. Calculate class mid-points (Col. 3) Mean = 40. and deviations from an arbitrarily 3. Multiply values of ‘d’ with chosen value, just like in the corresponding frequencies to get assumed mean method. In this ‘fd’ values (Col. 5). (note that the example, deviations have been total of this column is not zero taken from the value 40. (Col. 4) since deviations have been taken 2. Divide the deviations by a common from assumed mean). factor denoted as ‘C’. C = 5 in the 86 STATISTICS FOR ECONOMICS above example. The values so Set A 500 700 1000 obtained are ‘d'’ values (Col. 5). Set B 100000 120000 130000 3. Multiply ‘d'’ values with Suppose the values in Set A are corresponding ‘f'’ values (Col. 2) to the daily sales recorded by an ice- obtain ‘fd'’ values (Col. 6). cream vendor, while Set B has the daily sales of a big departmental store. 4. Multiply ‘fd'’ values with ‘d'’ values Range for Set A is 500 whereas for Set to get ‘fd'2’ values (Col. 7) B, it is 30,000. The value of Range is 5. Sum up values in Col. 6 and Col. much higher in Set B. Can you say 7 to get Σ fd' and Σ fd'2 values. that the variation in sales is higher for the departmental store? It can be 6. Apply the following formula. easily observed that the highest value 2 in Set A is double the smallest value, Sfd ¢ 2 Sfd ¢ s = - ·c whereas for the Set B, it is only 30% Sf Ł Sf ł higher. Thus absolute measures may 2 give misleading ideas about the extent 472 4 of variation specially when the or s = - ·5 40 Ł40 ł averages differ significantly. Another weakness of absolute or s = 11.8 - .01 · 5 measures is that they give the answer in the units in which original values s = 11.79 · 5 are expressed. Consequently, if the or s = 17.168 values are expressed in kilometers, the dispersion will also be in kilometers. Standard Deviation: Comments However, if the same values are Standard Deviation, the most widely expressed in meters, an absolute used measure of dispersion, is based measure will give the answer in meters on all values. Therefore a change in and the value of dispersion will appear even one value affects the value of standard deviation. It is independent to be 1000 times. of origin but not of scale. It is also To overcome these problems, useful in certain advanced statistical relative measures of dispersion can be problems. used. Each absolute measure has a relative counterpart. Thus, for Range, there is Coefficient of Range which is 5. ABSOLUTE AND RELATIVE MEASURES calculated as follows: OF DISPERSION L- S All the measures, described so far, are Coefficient of Range = absolute measures of dispersion. They L+ S calculate a value which, at times, is where L = Largest value difficult to interpret. For example, S = Smallest value consider the following two data sets: Similarly, for Quartile Deviation, it MEASURES OF DISPERSION 87 is Coefficient of Quartile Deviation be compared even across different which can be calculated as follows: groups having different units of Coefficient of Quartile Deviation measurement. Q3 - Q 1 7. LORENZ CURVE = rd Q3 + Q 1 where Q3=3 Quartile The measures of dispersion Q1 = 1st Quartile discussed so far give a numerical For Mean Deviation, it is value of dispersion. A graphical Coefficient of Mean Deviation. measure called Lorenz Curve is Coefficient of Mean Deviation = available for estimating dispersion. M.D.( x ) M.D.( Median ) You may have heard of statements like or ‘top 10% of the people of a country x Median Thus if Mean Deviation is earn 50% of the national income while calculated on the basis of the Mean, top 20% account for 80%’. An idea it is divided by the Mean. If Median is about income disparities is given by used to calculate Mean Deviation, it such figures. Lorenz Curve uses the is divided by the Median. information expressed in a cumulative For Standard Deviation, the manner to indicate the degree of relative measure is called Coefficient variability. It is specially useful in of Variation, calculated as below: comparing the variability of two or Coefficient of Variation more distributions. Given below are the monthly Standard Deviation incomes of employees of a company. = · 100 Arithmetic Mean TABLE 6.4 It is usually expressed in Incomes Number of employees percentage terms and is the most 0–5,000 5 commonly used relative measure of 5,000–10,000 10 dispersion. Since relative measures 10,000–20,000 18 are free from the units in which the 20,000–40,000 10 values have been expressed, they can 40,000–50,000 7 Example 16 Income Mid-points Cumulative Cumulative No. of Comulative Comulative limits mid-points mid-points as employees frequencies frequencies as percentages frequencies percentages (1) (2) (3) (4) (5) (6) (7) 0–5000 2500 2500 2.5 5 5 10 5000–10000 7500 10000 10.0 10 15 30 10000–20000 15000 25000 25.0 18 33 66 20000–40000 30000 55000 55.0 10 43 86 40000–50000 45000 100000 100.0 7 50 100 88 STATISTICS FOR ECONOMICS Construction of the Lorenz Curve from line OC has the highest Following steps are required. dispersion. 1. Calculate class mid-points and find cumulative totals as in Col. 3 in the example 16, given above. 2. Calculate cumulative frequencies as in Col. 6. 3. Express the grand totals of Col. 3 and 6 as 100, and convert the cumulative totals in these columns into percentages, as in Col. 4 and 7. 4. Now, on the graph paper, take the cumulative percentages of the variable (incomes) on Y axis and cumulative percentages of frequencies (number of employees) on X-axis, as in figure 6.1. Thus each axis will have values from ‘0’ to ‘100’. 5. Draw a line joining Co-ordinate 8. CONCLUSION (0, 0) with (100,100). This is called Although Range is the simplest to the line of equal distribution calculate and understand, it is unduly shown as line ‘OC’ in figure 6.1. affected by extreme values. QD is not 6. Plot the cumulative percentages of affected by extreme values as it is the variable with corresponding based on only middle 50% of the data. cumulative percentages of However, it is more dif ficult to frequency. Join these points to get interpret M.D. and S.D. both are based the curve OAC. upon deviations of values from their average. M.D. calculates average of Studying the Lorenz Curve deviations from the average but OC is called the line of equal ignores signs of deviations and distribution, since it would imply a therefore appears to be unmathema- situation like, top 20% people earn tical. Standard Deviation attempts to 20% of total income and top 60% earn calculate average deviation from 60% of the total income. The farther mean. Like M.D., it is based on all the curve OAC from this line, the values and is also applied in more greater is the variability present in the advanced statistical problems. It is distribution. If there are two or more the most widely used measure of curves, the one which is the farthest dispersion. MEASURES OF DISPERSION 89 Recap • A measure of dispersion improves our understanding about the behaviour of an economic variable. • Range and Quartile Deviation are based upon the spread of values. • M.D. and S.D. are based upon deviations of values from the average. • Measures of dispersion could be Absolute or Relative. • Absolute measures give the answer in the units in which data are expressed. • Relative smeasures are free from these units, and consequently can be used to compare different variables. • A graphic method, which estimates the dispersion from shape of a curve, is called Lorenz Curve. EXERCISES 1. A measure of dispersion is a good supplement to the central value in understanding a frequency distribution. Comment. 2. Which measure of dispersion is the best and how? 3. Some measures of dispersion depend upon the spread of values whereas some calculate the variation of values from a central value. Do you agree? 4. In a town, 25% of the persons earned more than Rs 45,000 whereas 75% earned more than 18,000. Calculate the absolute and relative values of dispersion. 5. The yield of wheat and rice per acre for 10 districts of a state is as under: District 1 2 3 4 5 6 7 8 9 10 Wheat 12 10 15 19 21 16 18 9 25 10 Rice 22 29 12 23 18 15 12 34 18 12 Calculate for each crop, (i) Range (ii) Q.D. (iii) Mean Deviation about Mean (iv) Mean Deviation about Median (v) Standard Deviation (vi) Which crop has greater variation? (vii) Compare the values of different measures for each crop. 6. In the previous question, calculate the relative measures of variation and indicate the value which, in your opinion, is more reliable. 7. A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are: 90 STATISTICS FOR ECONOMICS X 25 85 40 80 120 Y 50 70 65 45 80 Which batsman should be selected if we want, (i) a higher run getter, or (ii) a more reliable batsman in the team? 8. To check the quality of two brands of lightbulbs, their life in burning hours was estimated as under for 100 bulbs of each brand. Life No. of bulbs (in hrs) Brand A Brand B 0–50 15 2 50–100 20 8 100–150 18 60 150–200 25 25 200–250 22 5 100 100 (i) Which brand gives higher life? (ii) Which brand is more dependable? 9. Averge daily wage of 50 workers of a factory was Rs 200 with a Standard Deviation of Rs 40. Each worker is given a raise of Rs 20. What is the new average daily wage and standard deviation? Have the wages become more or less uniform? 10. If in the previous question, each worker is given a hike of 10 % in wages, how are the Mean and Standard Deviation values affected? 11. Calculate the Mean Deviation about Mean and Standard Deviation for the following distribution. Classes Frequencies 20–40 3 40–80 6 80–100 20 100–120 12 120–140 9 50 12. The sum of 10 values is 100 and the sum of their squares is 1090. Find the Coefficient of Variation. CHAPTER Index Numbers commodities have changed. Some Studying this chapter should items have become costlier, while enable you to: others have become cheaper. On his • understand the meaning of the return from the market, he tells his term index number; father about the change in price of the • become familiar with the use of each and every item, he bought. It is some widely used index bewildering to both. The industrial numbers; • calculate an index number; sector consists of many subsectors. • appreciate its limitations. Each of them is changing. The output of some subsectors are rising, while it is falling in some subsectors. The 1. INTRODUCTION changes are not uniform. Description You have learnt in the previous of the individual rates of change will chapters how summary measures can be difficult to understand. Can a be obtained from a mass of data. Now single figur e summarise these you will learn how to obtain summary changes? Look at the following cases: measures of change in a group of related variables. Case 1 Rabi goes to the market after a long An industrial worker was earning a gap. He finds that the prices of most salary of Rs 1,000 in 1982. Today, he 108 STATISTICS FOR ECONOMICS earns Rs 12,000. Can his standard of production in different sectors of an living be said to have risen 12 times industry, production of various during this period? By how much agricultural crops, cost of living etc. should his salary be raised so that he is as well off as before? Case 2 You must be reading about the sensex in the newspapers. The sensex crossing 8000 points is, indeed, greeted with euphoria. When, sensex dipped 600 points recently, it eroded investors’ wealth by Rs 1,53,690 crores. What exactly is sensex? Case 3 The government says inflation rate will Conventionally, index numbers are not accelerate due to the rise in the expressed in terms of percentage. Of price of petroleum products. How the two periods, the period with which does one measure inflation? the comparison is to be made, is These are a sample of questions known as the base period. The value you confront in your daily life. A study in the base period is given the index of the index number helps in number 100. If you want to know how analysing these questions. much the price has changed in 2005 from the level in 1990, then 1990 2. WHAT IS AN INDEX NUMBER becomes the base. The index number of any period is in proportion with it. An index number is a statistical device Thus an index number of 250 for measuring changes in the indicates that the value is two and half magnitude of a group of related variables. It represents the general times that of the base period. trend of diverging ratios, from which Price index numbers measure and it is calculated. It is a measure of the permit comparison of the prices of average change in a group of related certain goods. Quantity index variables over two different situations. numbers measure the changes in the The comparison may be between like physical volume of production, categories such as persons, schools, construction or employment. Though hospitals etc. An index number also price index numbers are more widely measures changes in the value of the used, a production index is also an variables such as prices of specified important indicator of the level of the list of commodities, volume of output in the economy. INDEX NUMBERS 109 3. CONSTRUCTION OF AN INDEX NUMBER The Aggregative Method In the following sections, the The formula for a simple aggregative principles of constructing an index price index is number will be illustrated through ΣP1 P01 = ¥ 100 price index numbers. ΣP0 Let us look at the following example: Where P1 and P0 indicate the price Example 1 of the commodity in the current period and base period respectively. Calculation of simple aggregative price Using the data from example 1, the index simple aggregative price index is TABLE 8.1 4+6+5+3 Commodity Base Current Percentage P01 = ¥ 100 = 138.5 period period change 2+5+4+2 price (Rs) price (Rs) Here, price is said to have risen by A 2 4 100 38.5 percent. B 5 6 20 Do you know that such an index C 4 5 25 is of limited use? The reason is that D 2 3 50 the units of measurement of prices of As you observe in this example, the various commodities are not the percentage changes are different for same. It is unweighted, because the relative importance of the items has every commodity. If the percentage not been properly reflected. The items changes were the same for all four ar e treated as having equal items, a single measure would have importance or weight. But what been sufficient to describe the change. happens in reality? In reality the items However, the percentage changes pur chased dif fer in order of differ and reporting the percentage importance. Food items occupy a change for every item will be large proportion of our expenditure. confusing. It happens when the In that case an equal rise in the price number of commodities is large, which of an item with large weight and that of an item with low weight will have is common in any r eal market different implications for the overall situation. A price index represents change in the price index. these changes by a single numerical The for mula for a weighted measure. aggregative price index is There are two methods of ΣP1q1 constructing an index number. It can P01 = ¥ 100 ΣP0 q1 be computed by the aggregative method and by the method of An index number becomes a weighted index when the relative averaging relatives. 110 STATISTICS FOR ECONOMICS importance of items is taken care of. 4 ¥ 10 + 6 ¥ 12 + 5 ¥ 20 + 3 ¥ 15 Here weights are quantity weights. To = ¥ 100 2 ¥ 10 + 5 ¥ 12 + 4 ¥ 20 + 2 ¥ 15 construct a weighted aggregative index, a well specified basket of 257 = ¥ 100 = 135.3 commodities is taken and its worth 190 each year is calculated. It thus This method uses the base period measures the changing value of a fixed quantities as weights. A weighted aggregate of goods. Since the total aggregative price index using base value changes with a fixed basket, the period quantities as weights, is also change is due to price change. known as Laspeyre’s price index. It Various methods of calculating a provides an explanation to the weighted aggregative index use question that if the expenditure on different baskets with respect to time. base period basket of commodities was Rs 100, how much should be the expenditure in the current period on the same basket of commodities? As you can see here, the value of base period quantities has risen by 35.3 per cent due to price rise. Using base period quantities as weights, the price is said to have risen by 35.3 percent. Since the current period quantities differ from the base period quantities, the index number using current period weights gives a different value of the index number. Example 2 ΣP1q1 Calculation of weighted aggregative P01 = ¥ 100 price index ΣP0 q1 TABLE 8.2 4 ¥ 5 + 6 ¥ 10 + 5 ¥ 15 + 3 ¥ 10 = ¥ 100 Base period Current period 2 ¥ 5 + 5 ¥ 10 + 4 ¥ 15 + 2 ¥ 15 Commodity Price Quantity Price Quality P0 q0 p1 q1 185 = ¥ 100 = 132.1 A 2 10 4 5 140 B 5 12 6 10 It uses the current period C 4 20 5 15 D 2 15 3 10 quantities as weights. A weighted aggregative price index using current ΣP1q1 period quantities as weights is known P01 = ¥ 100 as Paasche’s price index. It helps in ΣP0 q1 answering the question that, if the INDEX NUMBERS 111 the current period basket of The weighted index of price commodities was consumed in the relatives is the weighted arithmetic base period and if we were spending mean of price relatives defined as Rs 100 on it, how much should be the expenditure in current period on the ÊP ˆ ΣW Á 1 ¥ 100˜ same basket of commodities. A Ë P0 ¯ Paasche’s price index of 132.1 is P01 = ΣW interpreted as a price rise of 32.1 where W = Weight. percent. Using current period weights, In a weighted price relative index the price is said to have risen by 32.1 weights may be determined by the per cent. proportion or percentage of Method of Averaging relatives expenditure on them in total expenditure during the base period. When there is only one commodity, the It can also refer to the current period price index is the ratio of the price of depending on the formula used. These the commodity in the current period are, essentially, the value shares of to that in the base period, usually different commodities in the total expressed in percentage terms. The expenditure. In general the base method of averaging relatives takes period weight is preferred to the the average of these relatives when current period weight. It is because there are many commodities. The calculating the weight every year is price index number using price inconvenient. It also refers to the relatives is defined as changing values of different baskets. They are strictly not comparable. 1 p1 P01 = Σ ¥ 100 Example 3 shows the type of n p0 information one needs for calculating weighted price index. where P1 and Po indicate the price of the ith commodity in the current Example 3 period and base period respectively. Calculation of weighted price relatives The ratio (P1/P0) × 100 is also referred index to as price relative of the commodity. TABLE 8.3 n stands for the number of commodities. In the curr ent Commodity Base Current Price Weight year year price relative in % example price (in Rs) (in Rs.) 1 Ê 4 6 5 3ˆ P01 = Á + + + ˜ ¥ 100 = 149 A 2 4 200 40 4 Ë 2 5 4 2¯ B 5 6 120 30 C 4 5 125 20 Thus the prices of the commodities D 2 3 150 10 have risen by 49 percent. 112 STATISTICS FOR ECONOMICS The weighted price index is Consumer Price Index ÊP ˆ In India three CPI’s are constructed. ΣW Á 1 ¥ 100˜ They are CPI for industrial workers Ë P0 ¯ P01 = (1982 as base), CPI for urban non ΣW manual employees (1984–85 as base) and CPI for agricultural 40 ¥ 200 + 30 ¥ 120 + 20 ¥ 125 + 10 ¥ 150 = labourers (base 1986–87). They are 100 routinely calculated every month to = 156 analyse the impact of changes in the The weighted price index is 156. retail price on the cost of living of The price index has risen by 56 these three br oad categories of percent. The values of the unweighted consumers. The CPI for industrial price index and the weighted price workers and agricultural labourers index differ, as they should. The higher are published by Labour Bureau, rise in the weighted index is due to Shimla. The Central Statistical the doubling of the most important Organisation publishes the CPI number of urban non manual item A in example 3. employees. This is necessary because their typical consumption Activity baskets contain many dissimilar • Interchange the current period items. values with the base period The weight scheme in CPI for values, in the data given in industrial workers (1982=100) by example 2. Calculate the price major commodity groups is given index using Laspeyre’s, and in the following table. In this scheme Paasche’s for mula. What food has the largest weight. Food difference do you observe from being the most important category, the earlier illustration? any rise in the food price will have a significant impact on CPI. This also 4. SOME IMPORTANT INDEX NUMBERS explains the government’s frequent statement that oil price hike will not Consumer price index be inflationary. Consumer price index (CPI), also Major Group Weight in % known as the cost of living index, Food 57.00 measures the average change in retail Pan, supari, tobacco etc. 3.15 prices. The CPI for industrial workers Fuel & light 6.28 Housing 8.67 is increasingly considered the Clothing, bedding & footwear 8.54 appropriate indicator of general Misc. group 16.36 inflation, which shows the most General 100.00 accurate impact of price rise on the Source: Economic Survey, Government of cost of living of common people. India. Consider the statement that the CPI INDEX NUMBERS 113 for industrial workers(1982=100) is Wholesale price index 526 in January 2005. What does this The wholesale price index number statement mean? It means that if the indicates the change in the general industrial worker was spending Rs 100 in 1982 for a typical basket of price level. Unlike the CPI, it does not commodities, he needs Rs 526 in have any reference consumer January 2005 to be able to buy an category. It does not include items identical basket of commodities. It is pertaining to services like barber not necessary that he/she buys the charges, repairing etc. basket. What is important is whether What does the statement “WPI with he has the capability to buy it. 1993-94 as base is 189.1 in March, Example 4 2005” mean? It means that the general price level has risen by 89.1 Construction of consumer price index percent during this period. number. TABLE 8.4 Item Weight in % Base period Current period R=P1/P0 × 100 WR W price (Rs) price (Rs) (in%) Food 35 150 145 96.67 3883.45 Fuel 10 25 23 92.00 920.00 Cloth 20 75 65 86.67 1733.40 Rent 15 30 30 100.00 1500.00 Misc. 20 40 45 112.50 2250.00 9786.85 Industrial production index ΣWR 9786.85 CPI = = = 97.86 The index number of industrial ΣW 100 production measures changes in the level of industrial production This exercise shows that the cost comprising many industries. It of living has declined by 2.14 per cent. includes the production of the public What does an index larger than 100 and the private sector. It is a weighted average of quantity relatives. The indicate? It means a higher cost of formula for the index is living necessitating an upward Σq1 ¥ W adjustment in wages and salaries. The IIP01 = ¥ 100 rise is equal to the amount, it exceeds ΣW 100. If the index is 150, 50 percent In India, it is currently calculated every month with 1993–94 as the upward adjustment is required. The base. In table 8.6, you can see the salaries of the employees have to be index number of some industrial raised by 50 per cent. groupings along with their weights. 114 STATISTICS FOR ECONOMICS Wholesale Price Index these categories. Why does a compa- ratively lower performance of mining The commodity weights in the WPI and quarrying not pull down the are determined by the estimates of the commodity value of domestic general index? production and the value of imports inclusive of import duty during the Index number of agricultural base year. It is available on a weekly production basis. Commodities are broadly Index number of agricultural production classified into three categories viz is a weighted average of quantity primary articles, fuel, power, light and lubricants and manufactured relatives. Its base period is the products. The weight scheme is triennium ending 1981-82. In 2003– given below. The low weight of 04 the index number of agricultural fuel,power,light and lubricants production was 179.5. It means that explains how the government can agricultural production has increased get away with such a statement that by 79.5 percent over the average of the oil price hike will not be the three years 1979–80, 1980–81 and inflationary at least in the short run. 1981–82. Foodgrains have a weight of TABLE 8.5 62.92 percent in this index. Category Weight in % No. of items Primary articles 22.0 98 SENSEX Fuel, power, You ofen come across a news item in light & lubricants 14.2 19 Manufactured a newspaper, products 63.8 318 “Sensex breaches 8700 mark. BSE closes at 8650 points. Investor wealth Source: Economic Survey 2004–2005, rises by Rs 9,000 crore. The sensex Govt. of India, p–89 broke the 8700 mark for the first time in its history but ended off the mark TABLE 8.6 Broad industrial grouping and their at 8650, also a new record closing weights level”. Broad groupings Weight in % Index no. in The rise in sensex was at the May, 2005 highest level till date, which reflects Mining and the good health of the economy in quarrying 10.47 155.2 general. As the share prices increase, Manufacturing 79.36 222.7 reflected by the rise in sensex, the Electricity 10.17 196.7 value of wealth of the shareholders General index 213.0 also rises. As the table shows, the growth Look at another news item, performances of the broad industrial “Sensex dips 600 in 30 days flat. categories differ. The general index Rs 1,53,690 crore investor wealth represents the average performance of eroded. While the sensex has lost 338 INDEX NUMBERS 115 Bombay Stock Exchange Sensex is the short for m of Bombay Stock Exchange Sensitive Index with 1978–79 as base. The value of the sensex is with reference to this period. It is the benchmark index for the Indian stock market. It consists of 30 stocks which represent 13 sectors of the economy and the companies listed ar e leaders in their respective industries. If the sensex rises, it indicates that the market is doing well and investors expect better earnings from companies. It also indicates a gr owing confidence of investors in the basic health of the economy. points in two consecutive days, it has index number will replace wholesale eroded 6.8% or 598 points since price index. October 4 when it hit an all time high Producer Price Index at 8800 points. Investor wealth eroded by a staggering Rs 1,53,690 crore or Pr oducer price index number measures price changes from the 6.7% during the period.” producers’ perspective. It uses only It shows that all is not well with basic prices including taxes, trade the health of the economy. The margins and transport costs. A investors may find it hard to decide Working Gr oup on Revision of whether to invest or not. Wholesale Price Index (1993– 94=100) is inter alia examining the feasibility of switching over from WPI to a PPI in India as in many countries. 5. ISSUES IN THE CONSTRUCTION OF AN INDEX NUMBER You should keep certain important issues in mind, while constructing an index number. • You need to be clear about the Another useful index in recent purpose of the index. Calculation of a years is the human development volume index will be inappropriate, index. Very soon producers price when one needs a value index. 116 STATISTICS FOR ECONOMICS • Besides this, the items are not Activity equally important for different groups • Collect data from the local of consumers when a consumer price vegetable market over a week for, index is constructed. The rise in petrol at least 10 items. T ry to price may not directly impact the living construct the daily price index condition of the poor agricultural for the week. What problems do labourers. Thus the items to be you encounter in applying both methods for the construction of included in any index have to be a price index? selected carefully to be as representative as possible. Only then 6. INDEX NUMBER IN ECONOMICS you will get a meaningful picture of the change. Why do we need to use the index • Every index should have a base. numbers? Wholesale price index This base should be as normal as number (WPI), consumer price index possible. Extreme values should not number (CPI) and industrial be selected as base period. The period production index (IIP) are widely used should also not belong to too far in in policy making. • Consumer index number (CPI) or the past. The comparison between cost of living index numbers are 1993 and 2005 is much more helpful in wage negotiation, meaningful than a comparison formulation of income policy, price between 1960 and 2005. Many items policy, rent control, taxation and in a 1960 typical consumption basket general economic policy formulation. have disappeared at present. • The wholesale price index (WPI) is Therefore, the base year for any index used to eliminate the effect of changes number is routinely updated. in prices on aggregates such as • Another issue is the choice of the national income, capital formation etc. formula, which depends on the nature • The WPI is widely used to measure of question to be studied. The only the rate of inflation. Inflation is a difference between the Laspeyres’ general and continuing increase in index and Paasche’s index is the prices. If inflation becomes sufficiently weights used in these formulae. large, money may lose its traditional • Besides, there are many sources function as a medium of exchange and of data with different degrees of as a unit of account. Its primary reliability. Data of poor reliability will impact lies in lowering the value of give misleading results. Hence, due money. The weekly inflation rate is care should be taken in the collection given by of data. If primary data are not being Xt Xt used, then the most reliable source of 1 ¥ 100 where X and X secondary data should be chosen. X t -1 t t-1 INDEX NUMBERS 117 refer to the WPI for the t th and (t-1) • Sensex is a useful guide for th weeks. investors in the stock market. If the • CPI are used in calculating the sensex is rising, investors ar e purchasing power of money and real optimistic of the future performance wage: of the economy. It is an appropriate (i) Purchasing power of money = 1/ time for investment. Cost of living index (ii) Real wage = (Money wage/Cost of Where can we get these index living index) × 100 numbers? Some of the widely used index If the CPI (1982=100) is 526 in numbers are routinely published in January 2005 the equivalent of a the Economic Survey, an annual rupee in January, 2005 is given by publication of the Government of India are WPI, CPI, Index Number of Yield 100 Rs = 0.19 . It means that it is of Principal Crops, Index of Industrial 526 Production, Index of Foreign Trade. worth 19 paise in 1982. If the money wage of the consumer is Rs 10,000, Activity his real wage will be • Check from the newspapers and construct a time series of sensex 100 with 10 observations. What Rs 10, 000 ¥ = Rs 1, 901 happens when the base of the 526 consumer price index is shifted from 1982 to 2000? It means Rs 1,901 in 1982 has the same purchasing power as Rs 7. CONCLUSION 10,000 in January, 2005. If he/she Thus, the method of the index number was getting Rs 3,000 in 1982, he/ enables you to calculate a single she is worse off due to the rise in price. measure of change of a large number To maintain the 1982 standard of of items. Index numbers can be living the salary should be raised to calculated for price, quantity, volume Rs 15,780 obtained by multiplying the etc. base period salary by the factor 526/ 100. It is also clear from the formulae • Index of industrial production that the index numbers need to be gives us a quantitative figure about interpreted carefully. The items to be the change in production in the included and the choice of the base industrial sector. period are important. Index numbers • Agricultural production index are extremely important in policy provides us a ready reckoner of the making as is evident by their various performane of agricultural sector. uses. 118 STATISTICS FOR ECONOMICS Recap • An index number is a statistical device for measuring relative change in a large number of items. • There are several formulae for working out an index number and every formula needs to be interpreted carefully. • The choice of formula largely depends on the question of interest. • Widely used index numbers are wholesale price index, consumer price index, index of industrial production, agricultural production index and sensex. • The index numbers are indispensable in economic policy making. EXERCISES 1. An index number which accounts for the relative importance of the items is known as (i) weighted index (ii) simple aggregative index (iii) simple average of relatives 2. In most of the weighted index numbers the weight pertains to (i) base year (ii) current year (iii) both base and current year 3. The impact of change in the price of a commodity with little weight in the index will be (i) small (ii) large (iii) uncertain 4. A consumer price index measures changes in (i) retail prices (ii) wholesale prices (iii) producers prices 5. The item having the highest weight in consumer price index for industrial workers is (i) Food (ii) Housing (iii) Clothing 6. In general, inflation is calculated by using (i) wholesale price index (ii) consumer price index (iii) producers’ price index INDEX NUMBERS 119 7. Why do we need an index number? 8. What are the desirable properties of the base period? 9. Why is it essential to have different CPI for dif ferent categories of consumers? 10. What does a consumer price index for industrial workers measure? 11. What is the difference between a price index and a quantity index? 12. Is the change in any price reflected in a price index number? 13. Can the CPI number for urban non-manual employees represent the changes in the cost of living of the President of India? 14. The monthly per capita expenditure incurred by workers for an industrial centre during 1980 and 2005 on the following items are given below. The weights of these items are 75,10, 5, 6 and 4 respectively. Prepare a weighted index number for cost of living for 2005 with 1980 as the base. Items Price in 1980 Price in 2005 Food 100 200 Clothing 20 25 Fuel & lighting 15 20 House rent 30 40 Misc 35 65 15. Read the following table carefully and give your comments. INDEX OF INDUSTRIAL PRODUCTION BASE 1993–94 Industry Weight in % 1996–97 2003–2004 General index 100 130.8 189.0 Mining and quarrying 10.73 118.2 146.9 Manufacturing 79.58 133.6 196.6 Electricity 10.69 122.0 172.6 16. Try to list the important items of consumption in your family. 17. If the salary of a person in the base year is Rs 4,000 per annum and the current year salary is Rs 6,000, by how much should his salary rise to maintain the same standard of living if the CPI is 400? 18. The consumer price index for June, 2005 was 125. The food index was 120 and that of other items 135. What is the percentage of the total weight given to food? 19. An enquiry into the budgets of the middle class families in a certain city gave the following information; 120 STATISTICS FOR ECONOMICS Expenses on items Food Fuel Clothing Rent Misc. 35% 10% 20% 15% 20% Price (in Rs) in 2004 1500 250 750 300 400 Price (in Rs) in 1995 1400 200 500 200 250 What is the cost of living index of 2004 as compared with 1995? 20. Record the daily expenditure, quantities bought and prices paid per unit of the daily purchases of your family for two weeks. How has the price change affected your family? 21. Given the following data- Year CPI of industrial CPI of urban CPI of agricultural WPI workers non-manual labourers (1993–94=100) (1982 =100) employees (1986–87 = 100) (1984–85 = 100) 1995–96 313 257 234 121.6 1996–97 342 283 256 127.2 1997–98 366 302 264 132.8 1998–99 414 337 293 140.7 1999–00 428 352 306 145.3 2000–01 444 352 306 155.7 2001–02 463 390 309 161.3 2002–03 482 405 319 166.8 2003–04 500 420 331 175.9 Source: Economic Survey, Government of India.2004–2005 (i) Calculate the inflation rates using different index numbers. (ii) Comment on the relative values of the index numbers. (iii) Are they comparable? Activity • Consult your class teacher to make a list of widely used index numbers. Get the most recent data indicating the source. Can you tell what the unit of an index number is? • Make a table of consumer price index for industrial workers in the last 10 years and calculate the purchasing power of money. How is it changing? f m X f